Algebra II

Function Composition

Lesson

Function composition means applying one function to the result of another. The notation $f(g(x))$ reads as “f of g of x” and means: first do $g$ , then do $f$ on the result.

It can also be written using the small circle: $(f \circ g)(x) = f(g(x))$ .

To evaluate $f(g(x))$ at a specific number:

Plug the number into $g$ first.
Take that result and plug it into $f$ .

Worked example 1

Given $f(x) = x + 4$ and $g(x) = 2x$ , find $f(g(3))$ .

First, compute g(3):

g(3) = 2(3) = 6

Then plug 6 into f:

f(6) = 6 + 4 = 10

Worked example 2

With $f(x) = x^2$ and $g(x) = x - 3$ , find $g(f(5))$ .

Order matters! Inner first, then outer:

f(5) = 25

g(25) = 25 - 3 = 22

Note that $g(f(x))$ usually does NOT equal $f(g(x))$ .

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

f(x) = x + 1, g(x) = 2x; \text{ find } f(g(3))

Problem 2

f(x) = x + 4, g(x) = 2x; \text{ find } f(g(3))

Problem 3

f(x) = 3x, g(x) = x - 1; \text{ find } f(g(5))

Problem 4

f(x) = x - 2, g(x) = x + 5; \text{ find } f(g(0))

Practice

Standard problems matching the lesson.

Problem 5

f(x) = 2x + 1, g(x) = x - 3; \text{ find } f(g(4))

Problem 6

f(x) = x^2, g(x) = x - 3; \text{ find } g(f(5))

Problem 7

f(x) = x^2, g(x) = x + 1; \text{ find } f(g(2))

Problem 8

f(x) = 3x - 2, g(x) = x^2; \text{ find } f(g(2))

Problem 9

f(x) = x + 7, g(x) = 2x - 1; \text{ find } f(g(3))

Problem 10

f(x) = 4x, g(x) = x + 2; \text{ find } g(f(3))

Problem 11

f(x) = x^2 + 1, g(x) = x - 2; \text{ find } f(g(5))

Problem 12

f(x) = 2x, g(x) = x + 3; \text{ find } f(g(-1))

Problem 13

f(x) = x - 6, g(x) = 3x; \text{ find } f(g(4))

Problem 14

f(x) = -x + 2, g(x) = x - 5; \text{ find } f(g(8))

Problem 15

f(x) = x + 3, g(x) = x^2; \text{ find } g(f(2))

Problem 16

f(x) = 5x, g(x) = x - 4; \text{ find } f(g(10))

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 17

f(x) = x^2, g(x) = 2x + 1; \text{ find } f(g(-2))

Problem 18

f(x) = 3x - 4, g(x) = x^2 + 1; \text{ find } f(g(-1))

Problem 19

f(x) = 2x + 5, g(x) = x - 8; \text{ find } g(f(g(7)))

Problem 20

f(x) = -2x + 3, g(x) = 4 - x; \text{ find } f(g(-1))

Problem 21

f(x) = x^2 - x, g(x) = x + 2; \text{ find } f(g(-3))

Problem 22

f(x) = x + 1, g(x) = 2x - 3; \text{ find } f(g(f(4)))