Composition function worksheet - Free Printable

Let’s solve each problem step by step. The notation `(f ∘ g)(x)` means `f(g(x))`, and `(g ∘ f)(x)` means `g(f(x))`. We’ll evaluate each composite function at the given input.

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Problem 1

Given:
`f(x) = 3x - 5`
`g(x) = x²`

Find:
(a) `(f ∘ g)(3)` = `f(g(3))`
First, compute `g(3) = 3² = 9`
Then, `f(9) = 3*9 - 5 = 27 - 5 = 22`
✔ So, `(f ∘ g)(3) = 22`

(b) `(g ∘ f)(3)` = `g(f(3))`
First, compute `f(3) = 3*3 - 5 = 9 - 5 = 4`
Then, `g(4) = 4² = 16`
✔ So, `(g ∘ f)(3) = 16`

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Problem 2

Given:
`f(x) = -9x - 9`
`g(x) = √(x - 9)`

Find:
(a) `(f ∘ g)(10)` = `f(g(10))`
First, `g(10) = √(10 - 9) = √1 = 1`
Then, `f(1) = -9*1 - 9 = -18`
✔ So, `(f ∘ g)(10) = -18`

(b) `(f ∘ f)(-2)` = `f(f(-2))`
First, `f(-2) = -9*(-2) - 9 = 18 - 9 = 9`
Then, `f(9) = -9*9 - 9 = -81 - 9 = -90`
✔ So, `(f ∘ f)(-2) = -90`

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Problem 3

Given:
`f(x) = -4x + 2`
`g(x) = √(x - 8)`

Find:
(a) `(f ∘ g)(12)` = `f(g(12))`
First, `g(12) = √(12 - 8) = √4 = 2`
Then, `f(2) = -4*2 + 2 = -8 + 2 = -6`
✔ So, `(f ∘ g)(12) = -6`

(b) `(f ∘ f)(5)` = `f(f(5))`
First, `f(5) = -4*5 + 2 = -20 + 2 = -18`
Then, `f(-18) = -4*(-18) + 2 = 72 + 2 = 74`
✔ So, `(f ∘ f)(5) = 74`

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Problem 4

Given:
`f(x) = -3x + 4`
`g(x) = x²`

Find:
(a) `(g ∘ f)(-2)` = `g(f(-2))`
First, `f(-2) = -3*(-2) + 4 = 6 + 4 = 10`
Then, `g(10) = 10² = 100`
✔ So, `(g ∘ f)(-2) = 100`

(b) `(f ∘ g)(-2)` = `f(g(-2))`
First, `g(-2) = (-2)² = 4`
Then, `f(4) = -3*4 + 4 = -12 + 4 = -8`
✔ So, `(f ∘ g)(-2) = -8`

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Problem 5

Given:
`f(x) = -2x + 1`
`g(x) = √(x² - 5)`
`h(x) = 5x`

Find:
(a) `(h ∘ g ∘ f)(2)` = `h(g(f(2)))`
We work from the inside out:

Step 1: `f(2) = -2*2 + 1 = -4 + 1 = -3`
Step 2: `g(f(2)) = g(-3) = √[(-3)² - 5] = √[9 - 5] = √4 = 2`
Step 3: `h(g(f(2))) = h(2) = 5*2 = 10`
✔ So, `(h ∘ g ∘ f)(2) = 10`

(b) `f(4h(3))` — Note: this is NOT a composition, but function evaluation with nested input.

First, compute `h(3) = 5*3 = 15`
Then, `4h(3) = 4 * 15 = 60`
Now, `f(60) = -2*60 + 1 = -120 + 1 = -119`
✔ So, `f(4h(3)) = -119`

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## ✔ Final Answers:

1.
`(f ∘ g)(3) = 22`
`(g ∘ f)(3) = 16`

2.
`(f ∘ g)(10) = -18`
`(f ∘ f)(-2) = -90`

3.
`(f ∘ g)(12) = -6`
`(f ∘ f)(5) = 74`

4.
`(g ∘ f)(-2) = 100`
`(f ∘ g)(-2) = -8`

5.
`(h ∘ g ∘ f)(2) = 10`
`f(4h(3)) = -119`

Let me know if you’d like to see graphs or further explanations!