Composition of Functions SAT Notes

Function composition connects functions in a pipeline: the output of one function becomes the input of another. The notation $(f\circ g)(x)=f(g(x))$ is read as ‘f of g of x.’ It does not mean multiplication.

Learning objectives

Evaluate $$f(g(x))$$ from equations, tables, and ordered pairs.
Explain why $f\circ g$ and $g\circ f$ are generally different.
Substitute an entire expression into every occurrence of the outer input.
Check both domain conditions before declaring a composition defined.

Meaning and order

Composite function: A function formed by using the output of one function as the input of another. In $$f(g(x))$$ , $$g$$ is the inner function and $$f$$ is the outer function.

Composition order

\[(f\circ g)(x)=f(g(x)),\qquad (g\circ f)(x)=g(f(x))\]

Apply the function closest to $$x$$ first. Reversing the symbols reverses the pipeline and usually changes the result.

Apply the inner function firstA five-stage pipeline follows an input through function g and then function f to produce the composite output.

$x$
Start with the given input.
$g$
Apply the inner function first.
$g(x)$
Carry this entire result forward.
$f$
Use the inner output as the new input.
$f(g(x))$
This equals f composed with g at x.

Reversing the order changes the pipelineA five-stage pipeline follows an input through function f and then function g to show the reversed composition.

$x$
Start with the same input.
$f$
Apply f first in this order.
$f(x)$
Carry the complete result forward.
$g$
Use the inner output as g's input.
$g(f(x))$
This equals g composed with f at x.

Compose equations from the inside out

Reliable substitution procedure

Name the inner function
In $$f(g(x))$$ , evaluate or copy $$g(x)$$ first.
Open the outer rule
Write the formula for $$f(u)$$ , using a temporary input $$u$$ if helpful.
Replace every input
Substitute the entire expression $$g(x)$$ for every $$u$$ , using parentheses.
Simplify last
Expand and combine only after the substitution is structurally correct.

Worked example

Compare both orders

Let $$f(x)=2x+5$$ and $$g(x)=x^2-4$$ . Find both composites.

Compose f after g
$$f(g(x))=2(x^2-4)+5=2x^2-3$$ .
Compose g after f
$$g(f(x))=(2x+5)^2-4=4x^2+20x+21$$ .
Compare
The outputs have different coefficients and terms, so the compositions are not equal.

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

and

(g\circ f)(x)=4x^2+20x+21

Composition from tables and ordered pairs

Original function table for composition lookup
Input	f output	g output
$$-3$$	$$4$$	$$1$$
$$1$$	$$-2$$	$$5$$
$$4$$	$$0$$	$$-3$$
$$5$$	$$7$$	$$4$$

Worked example

Follow two table lookups

Use the table to find $$f(g(5))$$ and $$g(f(-3))$$ .

First composition
The table gives $$g(5)=4$$ , then $$f(4)=0$$ .
Second composition
The table gives $$f(-3)=4$$ , then $$g(4)=-3$$ .

\(f(g(5))=0\)

and

\(g(f(-3))=-3\)

The two domain conditions

Common mistakes and traps

Evaluating $$f$$ first in $$f(g(x))$$ .
Treating $f\circ g$ as the product $$f(x)g(x)$$ .
Substituting $$g(x)$$ into only one occurrence of the outer input.
Assuming $f\circ g=g\circ f$ without calculation.
Reading the right output from the wrong row of a table.
Using an inner output that is not in the outer function's domain.

Mini check

Check the inner-first rule

If $$f(x)=3x-1$$ and $$g(x)=x+4$$ , what is $$f(g(2))$$ ?

$$9$$
$$17$$
$$21$$
$$25$$

Show answer and explanation

Answer: $$17$$

First $$g(2)=6$$ . Then $$f(6)=3(6)-1=17$$ .

Key takeaways

What to remember

$(f\circ g)(x)$ means $$f(g(x))$$ : apply $$g$$ first.
Composition order is generally not interchangeable.
Equations, tables, and ordered pairs all use the same inner-then-outer logic.
A composition needs a valid input at both stages.

Continue learning

Put these notes into practice

Apply the ideas with SAT-style questions, then reinforce key details with flashcards.

Go to Practice Review Flashcards

Input	f output	g output
$\(-3\)$	$\(4\)$	$\(1\)$
$\(1\)$	$\(-2\)$	$\(5\)$
$\(4\)$	$\(0\)$	$\(-3\)$
$\(5\)$	$\(7\)$	$\(4\)$