Skill: Evaluate composite functions

Explanation and Free Practice Resources

Evaluating a composite function means finding the output of one function and using that value as the input to another, represented symbolically by $(f\circ g)(x)=f(g(x))$ ; the order of composition matters, and $f(g(x))$ is not multiplication. This understanding includes evaluating numerical and algebraic compositions, interpreting them from tables or graphs, and checking that intermediate outputs lie in the domain of the outer function, without extending to abstract or highly generalized composition theory.

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Detailed Explanation: Evaluate composite functions

To evaluate a composite function, work from the inside out:

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

This means:

Find $(g(x))$ .
Use that output as the input for $f$ .

The notation $(f(g(x)))$ does not mean $(f(x)\cdot g(x))$ .

Worked example

Let

f(x)=2x+3 \qquad\text{and}\qquad g(x)=x^2-1.

Find $((f\circ g)(2))$ .

Since

(f\circ g)(2)=f(g(2)),

start with the inside function, $g$ :

g(2)=2^2-1=4-1=3.

Now use this output, $3$ , as the input for $f$ :

f(3)=2(3)+3=6+3=9.

Therefore,

\boxed{(f\circ g)(2)=9}.

You can also find the rule for the composition:

(f\circ g)(x)=f(x^2-1) =2(x^2-1)+3 =2x^2+1.

The order matters. For example,

(g\circ f)(2)=g(f(2)).

Because $(f(2)=7)$ ,

g(7)=7^2-1=48,

so $((g\circ f)(2)=48)$ , not $9$ .

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Function Composition to Domain - Integer over Root of Quadratic (Real Roots) to Domain Definition

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