Combining Functions, Composition, and Inverses

Discussion prompt

Before we open Combining Functions, Composition, and Inverses: without looking back, what was the main idea of Quadratic Functions, Parabolas, and Optimization, and what could you do by the end of it that you could not do before?

Hint: One sentence for the idea, one for the skill. If the second one is blank, that is the part to revisit.

Discussion prompt

You already add, subtract, multiply, and divide numbers. You can do all four to functions too.

That is stated as though it always holds. Do one of two things: produce a case where it fails, or say precisely what rules such a case out. "It just does" is not on the menu.

Hint: Hunt at the extremes first — zero, one, negative, empty, equal. If every extreme survives, the reason they survive is the proof.

Discussion prompt

Explain Two vending machines, one dollar by analogy to something with no College Algebra in it at all — a queue, a recipe, a map, a bank balance, whatever fits. Then say where your analogy breaks.

Hint: An analogy that never breaks is not an analogy, it is the same idea wearing a hat. Find the seam — that is the part that is actually new.

Discussion prompt

Explain The four combinations to a student a year behind you. No notation, no jargon they have not met — and it still has to be true.

Hint: If your explanation needs a symbol they have never seen, you are describing the notation rather than the idea.

Put in order

Put the moves of Worked example: a sum and a difference into the order they have to happen.

1. Write the sum as the two rules added
2. Drop the parentheses and combine like terms
3. For the difference, subtract the whole second rule
4. Combine like terms
5. Verify at the test input five

Why: These are the moves of the worked example in the order it makes them, and each one is set up by the one before it. The definition says the sum function's rule is literally f of x plus g of x.

Animation

Shows: Restricting to make it invertible — a rendered Manim animation.

Rendered with Manim.

Takeaway: Keep only the right half and the square root becomes a genuine inverse.

Discussion prompt

Worked example: the product — before any calculation: what is the plan? Name the moves in order, in plain English, without doing the arithmetic.

Hint: It starts with: Write the product as the two rules multiplied

Animation

Shows: Each line of the worked example "the product", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: Multiplying the separate outputs gives 16 times 2, which is 32. The expanded rule gives 125 minus 75 minus 45 plus 27, which is also 32. They agree.

Predict first

Divide the same two functions and state the domain.

Commit before you compute: what does Worked example: the quotient and its domain come out to? A rough magnitude and the right form is enough — the point is to have something concrete to be wrong about.

Correct: Verify at the test input five, and test the excluded input

Why: A prediction you can defend turns the computation into a check rather than a leap of faith — and an answer that contradicts it is caught on the spot. At five the long way gives 16 divided by 2, which is 8, and the short rule gives 5 plus 3, which is 8.

Animation

Shows: Each line of the worked example "the quotient and its domain", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: At five the long way gives 16 divided by 2, which is 8, and the short rule gives 5 plus 3, which is 8. At three the original quotient is zero divided by zero, which is undefined - exactly why 3 is excluded.

Predict first

A student writes this, and it looks reasonable:

Simplify first, then read the domain from what is left.

It is wrong. Say what breaks — and say it before you turn the page.

Correct: The leftover rule is a polynomial, and polynomials accept everything - so it looks safe.

Find the restriction from the original denominator first, before any cancelling.

Why: The leftover rule is a polynomial, and polynomials accept everything - so it looks safe.

Annotate

From Trap: reading the domain off the simplified quotient — read this one piece at a time. What is each part doing?

On: \( \frac{x^2-9}{x-3} = x + 3 \)

• The leftover rule is a polynomial, and polynomials accept everything - so it looks safe.
• Substituting three into the original quotient gives zero over zero. A function cannot have a value there, so three was never in the domain.
• Cancelling changes the way the rule looks, but it cannot give the function a value it never had.

Put in order

These are the steps of Pattern: combining two functions, scrambled. Put them back in order before the next slide shows you.

1. Write out both rules in parentheses, then apply the operation.
2. For a difference, distribute the minus sign over every term of the second rule.
3. Simplify.

Why: This is the order the recipe itself gives. Recalling the sequence without the slide in front of you is the difference between recognising the method and being able to run it — most of what goes wrong in practice is a step done out of turn.

Eliminate the wrong options

What is the domain of the quotient function f divided by g?

3 of these 4 are wrong. Strike them one at a time, and say what rules each one out before you strike the next. The survivor is the answer.

• A. All real numbers except 1 and 3
• B. All real numbers except 3
• C. All real numbers except 1 and -1
• D. All real numbers

Survives elimination: A

Why: The bottom factors as (x - 1)(x - 3), so it is zero at x = 1 and at x = 3. Both inputs are thrown out before any simplifying, which leaves every real number except 1 and 3.

Check your understanding

What is the domain of the quotient function f divided by g?

• A. All real numbers except 1 and 3 (correct)
• B. All real numbers except 3
• C. All real numbers except 1 and -1
• D. All real numbers

Answer: A

Why: The bottom factors as (x - 1)(x - 3), so it is zero at x = 1 and at x = 3. Both inputs are thrown out before any simplifying, which leaves every real number except 1 and 3.

Why B tempts people

Simplified first: the (x - 1) factors cancel, leaving (x + 1)/(x - 3), and then the restriction at x = 1 was forgotten. A cancelled factor still excludes its value.

Why C tempts people

Used the zeros of the numerator instead of the denominator. A zero on top is a legal input - it just makes the output zero.

Why D tempts people

Treated the quotient like a sum or product, which has no extra restriction. Division adds the rule that the bottom cannot be zero.

Sort into buckets

Every line below is part of the definition of domain of a combination or of composition — one or the other, never both. Put each where it belongs.

domain of a combination

The intersection of the two domains; every input that both f and g accept.; If either machine rejects the input, the combination has nothing to compute.

composition

Applying one function to the result of another.; The inner function runs first, and whatever it returns becomes the input of the outer function.

b1

The intersection of the two domains: every input that both f and g accept. If either machine rejects the input, the combination has nothing to compute.

b2

Applying one function to the result of another. The inner function runs first, and whatever it returns becomes the input of the outer function.

Animation

Shows: The notation trap — a rendered Manim animation.

Rendered with Manim.

Takeaway: A superscript minus one means something different here than in algebra.

Discussion prompt

Evaluate the composition of these two functions at the input two.

What do you need to know — or decide — before the first line can be written? List everything the problem has to hand you.

Hint: Anything you would have to invent to get started is a thing the problem must supply.

Animation

Shows: Each line of the worked example "a composition at a number", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: Two enters the inner machine and leaves as one; one enters the outer machine and leaves as three. The chain closes, so the answer is three.

Predict first

A student writes this, and it looks reasonable:

Evaluate the composite at two, starting with whichever function's letter you read first.

It is wrong. Say what breaks — and say it before you turn the page.

Correct: English reads left to right, so the leftmost letter feels like it should go first.

The function sitting next to the input runs first. Read from the inside out.

Why: English reads left to right, so the leftmost letter feels like it should go first.

Comparison matrix

From Trap: composing in the wrong order: refill the value at two column from what you know. The rest of the table is as it appeared.

Animation

Shows: Composition is not commutative — a rendered Manim animation.

Rendered with Manim.

Takeaway: Read from the inside out, always.

Fill in the blanks

From Worked example: building the general composite rule — finish the line. Write what belongs on the right of the equals sign before you look.

f(\square) = 2\,\square + 1

Why: Producing the right-hand side unprompted is the difference between recognising this line and being able to use it. The outer machine doubles its input and adds one, no matter what that input is called.

Animation

Shows: Each line of the worked example "building the general composite rule", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: The step-by-step route gave three at the input two. The new rule gives two times four, minus five, which is also three. The general rule agrees with the specific computation.

Discussion prompt

Worked example: the other order, a different rule — before any calculation: what is the plan? Name the moves in order, in plain English, without doing the arithmetic.

Hint: It starts with: Write the outer rule, which is now g, with a blank

Animation

Shows: Each line of the worked example "the other order, a different rule", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: The rule gives sixteen plus eight minus two, which is twenty-two, matching the direct computation from the trap slide. The other order gave three, so these are genuinely different functions.

Predict first

The table runs: 1 | 3 | 4 · 2 | 1 | 3 · 3 | 4 | 1

In Worked example: composing from a table, given the rows so far: what is the next one — the row where x is 4?

Correct: 4 | 2 | 2

Why: The relationship between the columns, not the individual numbers, is what generates the next row. Read across the row for the input one and take the value in the g column.

Animation

Shows: Each line of the worked example "composing from a table", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: Reading each pair off the table in order confirms the two composites disagree at three of the four inputs. They only happen to agree at the input two.

Check your understanding

What is the value of the composite f of g of x at the input 2?

• A. 5 (correct)
• B. 11
• C. 10
• D. 7

Answer: A

Why: Run g first: g(2) = 3(2) - 4 = 2. Then run f on that output: f(2) = 2^2 + 1 = 5. The inner function's output, not the original input, is what f receives.

Why B tempts people

Composed in the wrong order: computed f(2) = 5 first, then g(5) = 3(5) - 4 = 11. That is the other composite, g of f.

Why C tempts people

Multiplied the two separate outputs, f(2) = 5 times g(2) = 2, giving 10. The composition circle is not a multiplication dot.

Why D tempts people

Added the two separate outputs, f(2) = 5 plus g(2) = 2, giving 7. That is the sum function evaluated at 2, not the composite.

Predict first

Build the composite and state its domain in interval notation.

Commit before you compute: what does Worked example: a composite domain, both gates come out to? A rough magnitude and the right form is enough — the point is to have something concrete to be wrong about.

Correct: Verify with legal inputs and with the excluded one

Why: A prediction you can defend turns the computation into a check rather than a leap of faith — and an answer that contradicts it is caught on the spot. At four the inner returns two and the outer returns one over negative one, which is negative one.

Animation

Shows: Each line of the worked example "a composite domain, both gates", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: At four the inner returns two and the outer returns one over negative one, which is negative one. At zero the inner returns zero and the outer returns negative one third. At nine the inner returns three, and the outer divides by zero, so nine truly must be thrown out.

Predict first

A student writes this, and it looks reasonable:

Simplify the composite first, then read the domain off whatever is left.

It is wrong. Say what breaks — and say it before you turn the page.

Correct: One divided by a fraction is that fraction turned upside down, so the whole thing collapses to a simple binomial.

Check the inner function's domain before simplifying anything.

Why: One divided by a fraction is that fraction turned upside down, so the whole thing collapses to a simple binomial.

Draw it

Translate both ways. First write the expression above as a sentence with no symbols in it at all. Then cover it, and write your sentence back as notation. If the two versions disagree, the disagreement is the thing to fix.

Put in order

Put the moves of Worked example: a composite hiding two restrictions into the order they have to happen.

1. Substitute the inner rule into the outer rule
2. Clear the little fraction by multiplying top and bottom by the inner denominator
3. Gate one: the inner function rejects zero
4. Gate two: the outer function rejects an input of one
5. Write the domain as three intervals
6. Verify at a legal input and at both excluded ones

Why: These are the moves of the worked example in the order it makes them, and each one is set up by the one before it. Wherever the outer rule refers to its input, write the entire inner rule instead.

Animation

Shows: Each line of the worked example "a composite hiding two restrictions", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: At three the inner gives two thirds, and the outer gives one over negative one third, which is negative three. The simplified rule gives three over negative one, also negative three. At zero the inner machine breaks; at two the inner returns one, which the outer machine refuses.

Predict first

What is the domain of the composite f of g of x?

Answer it in your own words, now, with nothing to choose from. The options are on the next slide — and picking the right one off a list is an easier skill than producing it.

Correct: [0, 25) union (25, infinity)

Why: The composite is 1 divided by the quantity (square root of x, minus 5). Gate one: the square root needs x greater than or equal to 0. Gate two: the outer rule fails when the square root equals 5, which happens at x = 25. So keep the ray from 0 up and puncture it at 25.

Check your understanding

What is the domain of the composite f of g of x?

• A. [0, 25) union (25, infinity) (correct)
• B. [0, infinity)
• C. (-infinity, 25) union (25, infinity)
• D. [0, 5) union (5, infinity)

Answer: A

Why: The composite is 1 divided by the quantity (square root of x, minus 5). Gate one: the square root needs x greater than or equal to 0. Gate two: the outer rule fails when the square root equals 5, which happens at x = 25. So keep the ray from 0 up and puncture it at 25.

Why B tempts people

Only checked gate one. The inner square root is happy at x = 25, but the outer rule then divides by zero, so 25 must still be removed.

Why C tempts people

Only checked gate two. The outer rule's restriction was found, but the square root still refuses every negative input, so the domain cannot include negatives.

Why D tempts people

Solved the gate-two equation without squaring: set the square root of x equal to 5 and reported x = 5 instead of x = 25. At x = 5 the square root is about 2.24, which the outer rule accepts fine.

Predict first

Worked example: decomposing a function is about to be worked. State your hypothesis first: which rule or definition decides this one, and what is the first move it forces? Then watch whether the example agrees with you.

Correct: Ask what you would compute first for a specific input

Why: You would work out three times the input plus seven before touching the exponent. That inside expression is the inner function.

A hypothesis you wrote down is falsifiable; a vague sense of how it will go is not. If the example opens somewhere else, that gap is the thing worth chasing.

Animation

Shows: Breaking a function into pieces — a rendered Manim animation.

Rendered with Manim.

Takeaway: The skill calculus will need for the chain rule.

Check your understanding

Which pair of functions satisfies h(x) = f(g(x)) for this h?

• A. f(x) = sqrt(x) and g(x) = 5x - 2 (correct)
• B. f(x) = 5x - 2 and g(x) = sqrt(x)
• C. f(x) = sqrt(x) - 2 and g(x) = 5x
• D. f(x) = sqrt(5x) and g(x) = x - 2

Answer: A

Why: The inside expression 5x - 2 is what you compute first, so it is the inner function g. The last action is taking the square root, so the outer function f is the square root. Composing gives the square root of (5x - 2), which is exactly h.

Why B tempts people

Reversed the roles: this composition gives 5 times the square root of x, minus 2, which is a different function entirely.

Why C tempts people

Put the subtraction in the outer rule. This composes to the square root of 5x, then subtracts 2, leaving the 2 outside the radical instead of inside.

Why D tempts people

Composes to the square root of 5 times the quantity x minus 2, which expands to the square root of 5x - 10, not 5x - 2.

Match the pairs

Match each term to the definition this lesson gave it — not the one you would guess from the word.

• t1. domain of a combination
• t2. composition
• t3. domain of a composite
• t4. decomposition
• t5. one-to-one function

• d1. The intersection of the two domains: every input that both f and g accept. If either machine rejects the input, the combination has nothing to compute.
• d2. Applying one function to the result of another. The inner function runs first, and whatever it returns becomes the input of the outer function.
• d3. Every input the inner function accepts, minus the inputs whose inner output the outer function would reject.
• d4. Given one complicated rule, naming an inner function and an outer function whose composition rebuilds it.
• d5. A function in which different inputs always produce different outputs. No output value is ever produced by two different inputs.

Why: These are the working definitions of domain of a combination, composition, domain of a composite, decomposition, one-to-one function as Combining Functions, Composition, and Inverses uses them. Pairing them correctly is the test of whether you could state each one with the slide switched off.

Discussion prompt

Read the picture before the words. What is this showing, and what is the one thing it is built to make obvious? Commit to an answer, then read on.

Hint: Name the parts, then say what changes between them — and if nothing changes, say what is being held still.

Sort into buckets

These are the pieces of Combining Functions, Composition, and Inverses, out of order. Put each one back under the part of the lesson it belongs to.

Combining Two Functions

Two functions, one input; Two vending machines, one dollar; The four combinations

Composition: Machines in Series

Composition: the output becomes the next input; An assembly line; The notation

One-to-One and Inverse Functions

One-to-one: no output is ever reused; Every seat has exactly one ticket holder; The horizontal line test

s1

Combining Two Functions is where Combining Functions, Composition, and Inverses puts Two functions, one input, Two vending machines, one dollar, The four combinations. Knowing which part of the lesson a problem belongs to is most of knowing which method to reach for.

s2

Composition: Machines in Series is where Combining Functions, Composition, and Inverses puts Composition: the output becomes the next input, An assembly line, The notation. Knowing which part of the lesson a problem belongs to is most of knowing which method to reach for.

s3

One-to-One and Inverse Functions is where Combining Functions, Composition, and Inverses puts One-to-one: no output is ever reused, Every seat has exactly one ticket holder, The horizontal line test. Knowing which part of the lesson a problem belongs to is most of knowing which method to reach for.

Animation

Shows: Which functions have inverses — a rendered Manim animation.

Rendered with Manim.

Takeaway: This fails the horizontal line test, so it has no inverse without restricting the domain.

Animation

Shows: Composing, carefully — a rendered Manim animation.

Rendered with Manim.

Takeaway: Two different functions, from the same two pieces.

Predict first

Which of these functions is one-to-one on its natural domain?

Commit to an answer, then rate it — certain, fairly sure, or guessing — and write the rating down before you turn the page.

Correct: h(x) = x^3 + 2

Why: Cubing preserves order, so the graph rises without ever turning around and every horizontal line meets it exactly once. If two cubes are equal then the two inputs are equal.

The rating matters as much as the answer: confident-and-wrong is the combination that survives revision, because nothing about it feels like it needs revisiting.

Check your understanding

Which of these functions is one-to-one on its natural domain?

• A. h(x) = x^3 + 2 (correct)
• B. h(x) = x^2 + 2
• C. h(x) = |x - 4|
• D. h(x) = x^4

Answer: A

Why: Cubing preserves order, so the graph rises without ever turning around and every horizontal line meets it exactly once. If two cubes are equal then the two inputs are equal.

Why B tempts people

Squaring makes opposite inputs agree: h(1) and h(-1) are both 3, so a horizontal line at 3 cuts the parabola twice.

Why C tempts people

Absolute value folds the number line at 4: h(3) and h(5) are both 1, so two different inputs share an output.

Why D tempts people

A fourth power also erases the sign: h(2) and h(-2) are both 16, so the graph fails the horizontal line test.

Animation

Shows: An inverse undoes the function — a rendered Manim animation.

Rendered with Manim.

Takeaway: Both compositions must give x back, or it is not an inverse.

Predict first

A student writes this, and it looks reasonable:

Treat the raised negative one as an ordinary exponent and flip the rule over.

It is wrong. Say what breaks — and say it before you turn the page.

Correct: A raised negative one on a number does mean flip it, so flipping the rule feels perfectly reasonable.

Ask what the rule actually does, then undo that action.

Why: A raised negative one on a number does mean flip it, so flipping the rule feels perfectly reasonable.

Annotate

From Trap: reading the inverse notation as a reciprocal — read this one piece at a time. What is each part doing?

On: \( f\big(f^{-1}(x)\big) = (x - 4) + 4 = x \qquad f^{-1}\big(f(x)\big) = (x + 4) - 4 = x \)

• A raised negative one on a number does mean flip it, so flipping the rule feels perfectly reasonable.
• The original sends three to seven, so a genuine inverse has to send seven back to three. This one sends seven to one eleventh, which is nowhere near three.
• There is one operation to undo, and undoing an addition means subtracting. Nothing gets flipped over.

Discussion prompt

Worked example: verifying a pair really are inverses — before any calculation: what is the plan? Name the moves in order, in plain English, without doing the arithmetic.

Hint: It starts with: Compose in the first direction

Animation

Shows: Each line of the worked example "verifying a pair really are inverses", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: The first function sends two to nine, and the second sends nine back to two, closing the loop. Both symbolic directions and the numeric round trip agree, so the two functions are inverses.

Predict first

For f(x) = 4x + 1, which function g satisfies the round-trip property in both directions?

Answer it in your own words, now, with nothing to choose from. The options are on the next slide — and picking the right one off a list is an easier skill than producing it.

Correct: g(x) = (x - 1)/4

Why: Composing gives f(g(x)) = 4 times (x - 1)/4, plus 1, which is (x - 1) + 1 = x, and g(f(x)) = ((4x + 1) - 1)/4 = 4x/4 = x. Both directions return the input, so these are inverses.

Check your understanding

For f(x) = 4x + 1, which function g satisfies the round-trip property in both directions?

• A. g(x) = (x - 1)/4 (correct)
• B. g(x) = 1/(4x + 1)
• C. g(x) = (x + 1)/4
• D. g(x) = 4x - 1

Answer: A

Why: Composing gives f(g(x)) = 4 times (x - 1)/4, plus 1, which is (x - 1) + 1 = x, and g(f(x)) = ((4x + 1) - 1)/4 = 4x/4 = x. Both directions return the input, so these are inverses.

Why B tempts people

Read the inverse notation as a reciprocal. Check it: f(1) = 5, but this rule sends 5 to 1/21, not back to 1.

Why C tempts people

Undid the added 1 with the wrong sign. Check it: f(1) = 5, and this rule sends 5 to 6/4, not back to 1.

Why D tempts people

Negated the constant but left the multiplication by 4 in place instead of undoing it. Check it: f(1) = 5, and this rule sends 5 to 19.

Animation

Shows: Finding an inverse — a rendered Manim animation.

Rendered with Manim.

Takeaway: Swap the variables, then solve for the new y.

Put in order

These are the steps of Pattern: find an inverse in four moves, scrambled. Put them back in order before the next slide shows you.

1. Replace the function name with the letter y.
2. Swap every x and every y.
3. Solve the new equation for y, undoing the operations in reverse order.
4. Rename the result as the inverse function, and state its domain.

Why: This is the order the recipe itself gives. Recalling the sequence without the slide in front of you is the difference between recognising the method and being able to run it — most of what goes wrong in practice is a step done out of turn.

Discussion prompt

Pattern: find an inverse in four moves works on the cases you have just seen. Push it to the edge: what is the most degenerate input it still handles — empty, zero, one item, everything equal — and what is the first case where it stops being true? Name the case, not just "it breaks".

Hint: Try the smallest legal input, then the largest, then the one where two things collide. Methods are specified at their edges; the middle takes care of itself.

Fill in the blanks

From Worked example: the inverse of a linear function — finish the line. Write what belongs on the right of the equals sign before you look.

f(x) = 3x - 7

Why: Producing the right-hand side unprompted is the difference between recognising this line and being able to use it. Nothing changes mathematically.

Animation

Shows: Each line of the worked example "the inverse of a linear function", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: Each composition collapses to the bare input, so this really is the inverse. Numerically, the original sends five to eight and the inverse sends eight back to five.

Predict first

A student writes this, and it looks reasonable:

Try to find the inverse without ever trading the letters.

It is wrong. Say what breaks — and say it before you turn the page.

Correct: The equation already has y by itself, so it feels like the work is finished before it started.

Swap the letters first. That one move is what reverses the machine.

Why: The equation already has y by itself, so it feels like the work is finished before it started.

Sort into buckets

Some of these hold up and some are the exact mistakes this lesson is built to prevent. Sort them.

Holds up

You already add, subtract, multiply, and divide numbers. You can do all four to functions too.; Picture two vending machines standing side by side. You put the same dollar amount into each one.; Read every one of these left to right as an instruction: evaluate both functions, then combine.

Breaks

Simplify first, then read the domain from what is left.; Evaluate the composite at two, starting with whichever function's letter you read first.

sound

These are stated as this lesson states them — each one survives the edge cases Combining Functions, Composition, and Inverses puts it through.

flawed

Each of these is lifted from a trap in this deck: reasonable-sounding, and wrong in a way that only shows up once you rely on it.

Discussion prompt

Find the inverse and state the restriction on each function.

What do you need to know — or decide — before the first line can be written? List everything the problem has to hand you.

Hint: Anything you would have to invent to get started is a thing the problem must supply.

Animation

Shows: An inverse is a reflection in y = x — a rendered Manim animation.

Rendered with Manim.

Takeaway: The graph and its inverse are mirror images across the diagonal.

Check your understanding

What is the inverse of this function?

• A. f inverse of x = 4x/(1 - x) (correct)
• B. f inverse of x = (x + 4)/x
• C. f inverse of x = (1 - x)/(4x)
• D. f inverse of x = 4x/(x - 1)

Answer: A

Why: Swapping gives x = y/(y + 4), so x(y + 4) = y, then xy + 4x = y, then 4x = y - xy = y(1 - x), so y = 4x/(1 - x). Numeric check: f(2) = 2/6 = 1/3, and 4(1/3)/(1 - 1/3) = (4/3)/(2/3) = 2.

Why B tempts people

Turned the original fraction upside down instead of swapping the letters and solving. That is the reciprocal of f, not its inverse.

Why C tempts people

Took the reciprocal of the correct answer, treating the raised negative one in the notation as an exponent.

Why D tempts people

Sign slip when factoring: wrote y(x - 1) instead of y(1 - x) after moving xy across, which flips the sign of the whole answer.

Animation

Shows: Domain and range trade places — a rendered Manim animation.

Rendered with Manim.

Takeaway: Which follows immediately from swapping x and y.

Predict first

Find the inverse, and state the domain of the original and of the inverse.

Commit before you compute: what does Worked example: a square root, its inverse, and both domains come out to? A rough magnitude and the right form is enough — the point is to have something concrete to be wrong about.

Correct: Verify both directions and one number

Why: A prediction you can defend turns the computation into a check rather than a leap of faith — and an answer that contradicts it is caught on the spot. Composing the inverse after the original squares the radical away and adds the two back, returning x for every input at least two.

Discussion prompt

Explain The two graphs are mirror images to a student a year behind you. No notation, no jargon they have not met — and it still has to be true.

Hint: If your explanation needs a symbol they have never seen, you are describing the notation rather than the idea.

Discussion prompt

Explain Restricting the domain to force one-to-one by analogy to something with no College Algebra in it at all — a queue, a recipe, a map, a bank balance, whatever fits. Then say where your analogy breaks.

Hint: An analogy that never breaks is not an analogy, it is the same idea wearing a hat. Find the seam — that is the part that is actually new.

Animation

Shows: The domain of a composition — a rendered Manim animation.

Rendered with Manim.

Takeaway: Both conditions, not just the outer one.

Discussion prompt

Worked example: inverting a parabola on a restricted domain — before any calculation: what is the plan? Name the moves in order, in plain English, without doing the arithmetic.

Hint: It starts with: Check that the restriction really makes it one-to-one

Animation

Shows: Each line of the worked example "inverting a parabola on a restricted domain", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: From three upward the graph is only the right-hand half of the parabola, which climbs steadily and never revisits an output. Without the restriction, three plus a number and three minus that same number would share an output.

Discussion prompt

A formula usually answers one direction of a question. Its inverse answers the other direction.

That is stated as though it always holds. Do one of two things: produce a case where it fails, or say precisely what rules such a case out. "It just does" is not on the menu.

Hint: Hunt at the extremes first — zero, one, negative, empty, equal. If every extreme survives, the reason they survive is the proof.

Put in order

Put the moves of Worked example: a conversion formula, run backwards into the order they have to happen.

1. Name the input and the output before touching the algebra
2. Undo the multiplication by five ninths
3. Undo the subtraction
4. Verify by round-tripping a temperature you already know

Why: These are the moves of the worked example in the order it makes them, and each one is set up by the one before it. Here the input is a Fahrenheit reading and the output is a Celsius reading, so the inverse must accept Celsius and return Fahrenheit.

Animation

Shows: Each line of the worked example "a conversion formula, run backwards", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: Water boils at 212 degrees Fahrenheit. The original formula gives five ninths of 180, which is 100 degrees Celsius. The new formula sends 100 back to nine fifths of 100, plus 32, which is 212. The loop closes, so the inverse is correct.

Eliminate the wrong options

Which formula gives the number of minutes used from a known bill?

3 of these 4 are wrong. Strike them one at a time, and say what rules each one out before you strike the next. The survivor is the answer.

• A. m = 20(c - 20)
• B. m = 20c - 20
• C. m = (c + 20)/0.05
• D. m = 1/(0.05c + 20)

Survives elimination: A

Why: Subtract the 20 dollar flat fee first, then divide by 0.05, which is the same as multiplying by 20. Check a bill of 35 dollars: 20 times 15 is 300 minutes, and 0.05 times 300 plus 20 gives 35 dollars back.

Check your understanding

Which formula gives the number of minutes used from a known bill?

• A. m = 20(c - 20) (correct)
• B. m = 20c - 20
• C. m = (c + 20)/0.05
• D. m = 1/(0.05c + 20)

Answer: A

Why: Subtract the 20 dollar flat fee first, then divide by 0.05, which is the same as multiplying by 20. Check a bill of 35 dollars: 20 times 15 is 300 minutes, and 0.05 times 300 plus 20 gives 35 dollars back.

Why B tempts people

Divided by 0.05 before subtracting the flat fee, so the 20 dollar charge got scaled up too. A 35 dollar bill would give 680 minutes, which actually costs 54 dollars.

Why C tempts people

Added the flat fee instead of removing it. A 35 dollar bill would give 1100 minutes, which costs 75 dollars.

Why D tempts people

Read the inverse notation as a reciprocal and flipped the whole formula over. The result is a tiny fraction of a minute, which cannot be a number of minutes.

Discussion prompt

Run Pattern: the whole toolkit on one page with this step confiscated:

Composing: substitute the inner rule into the outer rule, then simplify; check both domain gates.

Is it still possible? If it is, say what takes its place and what it costs you. If it is not, say exactly what that step was providing that nothing else does.

Hint: A step you can drop for free was never load-bearing. If you cannot drop it, name the thing that goes wrong the moment it is gone.

Discussion prompt

Outside this lesson: where does Combining Functions, Composition, and Inverses actually turn up? Name one concrete situation — a job, a piece of software someone ships, a decision somebody has to make — and say which part of Pattern: the whole toolkit on one page is doing the work in it.

Hint: Vague is the failure mode here. "Engineering" is not a situation; "deciding whether this build is fast enough to ship" is.

Draw it

One page, no notation unless you need it: draw how these connect — Combining Two Functions · Composition: Machines in Series · One-to-One and Inverse Functions. Put an arrow wherever one of them is what makes another possible, and label the arrow with why.