Composition of Functions

Unit 2 · Exponential and Logarithmic Functions

What AP Precalc asks here

The composition $(f \circ g)(x) = f(g(x))$ applies $g$ to $x$ first, then $f$ to the result. Composition is generally not commutative — $(f \circ g)(x) \neq (g \circ f)(x)$ . The domain has two requirements: $x$ must be in the domain of $g$ , and $g(x)$ must be in the domain of $f$ .

Composition

Definition

(f \circ g)(x) = f(g(x))

Three-function

(f \circ g \circ h)(x) = f(g(h(x)))

f \circ g

Both must hold

x \in \text{dom}(g) \;\text{ AND }\; g(x) \in \text{dom}(f)

Caution: Order matters.

(f \circ g)(x) = f(g(x))

, applying

g

first;

(g \circ f)(x) = g(f(x))

, applying

f

first. These usually differ.

Type 1

Evaluate a composition at a value

Compute the inner function first, then the outer. Substitute carefully.

Example 1

f(x) = 2x + 1

and

g(x) = x^2

, find

(f \circ g)(3)

. (A)

7

(B)

19

(C)

37

(D)

49

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Type 2

Express (f ∘ g)(x) as a formula

Substitute $g(x)$ into every $x$ in $f$ , then simplify.

Example 2

For

f(x) = 3x

and

g(x) = x + 2

, find

(f \circ g)(x)

. (A)

3x + 2

(B)

3x + 6

(C)

x + 6

(D)

3(x + 2)^2

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Type 3

Domain of a composition

Stack the restrictions: $x$ must be in dom( $g$ ), AND $g(x)$ must be in dom( $f$ ).

Example 3

For

f(x) = \dfrac{1}{x}

and

g(x) = \sqrt{x}

, find the domain of

(f \circ g)(x)

. (A)

\mathbb{R}

(B)

[0, \infty)

(C)

(0, \infty)

(D)

(-\infty, 0)

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Composition of Functions

What AP Precalc asks here

Composition

Domain of f∘gf \circ gf∘g

Evaluate a composition at a value

Express (f ∘ g)(x) as a formula

Domain of a composition

Domain of $f \circ g$