Function Operations

Function Inputs

f(x) =

g(x) =

Operation

(f+g)(x) = f(x) + g(x)

Example Pairs

Enter expressions using ^ for powers, sqrt() for radicals, * for multiplication. Coefficients like 2x are fine.

Results

Enter f(x) and g(x) above and press Compute All Operations to see all five results.

(f+g)(x) —

(f−g)(x) —

(f·g)(x) —

(f/g)(x) —

(f∘g)(x) —

Result

—

Domain

—

Operation Definitions

Notation	Definition	Domain Restriction
(f+g)(x)	f(x) + g(x)	Dom(f) ∩ Dom(g)
(f−g)(x)	f(x) − g(x)	Dom(f) ∩ Dom(g)
(f·g)(x)	f(x) · g(x)	Dom(f) ∩ Dom(g)
(f/g)(x)	f(x) / g(x)	Dom(f) ∩ Dom(g), g(x) ≠ 0
(f∘g)(x)	f(g(x))	x in Dom(g) and g(x) in Dom(f)

For the sum, difference, and product operations, the domain is just the intersection of both individual domains — all x values where both functions are defined.

Composition: f∘g ≠ g∘f in General

(f∘g)(x) = f(g(x)) — substitute g into f

Order matters: (f∘g)(x) means apply g first, then feed the result into f. (g∘f)(x) does the opposite — so they usually give different answers.

Domain of f∘g: x must be in the domain of g, AND the output g(x) must land inside the domain of f. This means f∘g can have a smaller domain than either function alone.

Composition with polynomials: Substitute the entire expression for g(x) in place of every x in f(x), then expand and collect like terms.

Write out g(x) in parentheses before substituting.
Expand powers carefully: (ax+b)² = a²x²+2abx+b².
Collect like terms after every substitution.
For (f/g): always state that g(x) ≠ 0 in the domain.

Function operations giving you trouble?