Composition of functions

About this lesson

Precalculus › Functions and their graphs

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Composition of functions

Warm-up

Three quick questions from earlier lessons.

Discover

gg doubles a number: g(x)=2xg(x) = 2x. ff adds 1: f(x)=x+1f(x) = x + 1. Start with 3: double it, then add 1. You have computed f(g(3))f(g(3)). What is it?

  1. 7
  2. 8
  3. 24

Key idea

Composing means using the output of one function as the input of another.

f(g(x))f(g(x)): first apply gg to xx, then apply ff to the result. Work from the inside out. In general f(g(x))≠g(f(x))f(g(x)) \ne g(f(x)).

Worked example

f(x)=x2f(x) = x^2 and g(x)=x−3g(x) = x - 3. Find f(g(5))f(g(5)), then f(g(x))f(g(x)).

g(5)=5−3=2g(5) = 5 - 3 = 2
The inside first.
f(g(5))=f(2)=4f(g(5)) = f(2) = 4
Then f of that result.
f(g(x))=(x−3)2f(g(x)) = (x - 3)^2
With x: put g(x) = x − 3 in place of x in f.

Your turn, guided

Your turn: f(x)=2x+1f(x) = 2x + 1 and g(x)=x2g(x) = x^2. Find f(g(3))f(g(3)).

g(3)=9g(3) = 9
The inside first.
f(g(3))=f(9)=19f(g(3)) = f(9) = 19
Then 2 · 9 + 1.

Practice ★

  1. f(x)=2x+2f(x) = 2x + 2 and g(x)=x2−2g(x) = x^2 - 2. Find f(g(3))f(g(3)).
  2. f(x)=3x+3f(x) = 3x + 3 and g(x)=x2−4g(x) = x^2 - 4. Find f(g(3))f(g(3)).
  3. f(x)=4x+3f(x) = 4x + 3 and g(x)=x2−1g(x) = x^2 - 1. Find f(g(2))f(g(2)).
  4. f(x)=3x+1f(x) = 3x + 1 and g(x)=x2−5g(x) = x^2 - 5. Find f(g(3))f(g(3)).

Watch out!

Practice ★★

  1. f(x)=6x+6f(x) = 6x + 6 and g(x)=x+5g(x) = x + 5. Write f(g(x))f(g(x)).
  2. f(x)=5x+7f(x) = 5x + 7 and g(x)=x+2g(x) = x + 2. Write f(g(x))f(g(x)).
  3. f(x)=6x+8f(x) = 6x + 8 and g(x)=x+3g(x) = x + 3. Write f(g(x))f(g(x)).
  1. g(1)=5g(1) = 5, g(5)=7g(5) = 7, f(1)=7f(1) = 7 and f(5)=1f(5) = 1. Find f(g(1))f(g(1)).
  2. g(2)=6g(2) = 6, g(6)=7g(6) = 7, f(2)=7f(2) = 7 and f(6)=2f(6) = 2. Find f(g(2))f(g(2)).

Exit ticket

Three questions to check you have got it.

  1. f(x)=2x+2f(x) = 2x + 2 and g(x)=x2−2g(x) = x^2 - 2. Find f(g(3))f(g(3)).
  2. f(x)=3x+3f(x) = 3x + 3 and g(x)=x2−4g(x) = x^2 - 4. Find f(g(3))f(g(3)).
  3. f(x)=4x+3f(x) = 4x + 3 and g(x)=x2−1g(x) = x^2 - 1. Find f(g(2))f(g(2)).

Summary card

Composition

  • f(g(x))f(g(x)): apply gg first, then ff.
  • Work from the inside out.
  • The order matters: f(g(x))f(g(x)) and g(f(x))g(f(x)) usually differ.

Watch out: f(g(x))f(g(x)) is not f(x)⋅g(x)f(x) \cdot g(x).

End of the lesson

Next lesson: Transformations: shifts and reflections Back to the chapter

Answers: Composition of functions

A

  1. 1616
  2. 1818
  3. 1515
  4. 1313

B

  1. 6x+366 x + 36
  2. 5x+175 x + 17
  3. 6x+266 x + 26

C

  1. 11
  2. 22

D

  1. 1616
  2. 1818
  3. 1515

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