Solved Composition of Functions Worksheet Let f(x) = 2x - 1 ... - Free Printable

Let's solve the Composition of Functions Worksheet step by step.

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Part I:

Given:
- $ f(x) = 2x - 1 $
- $ g(x) = 3x $
- $ h(x) = x^2 + 1 $

We will compute each composite function by substituting values appropriately.

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#### 1. $ f(g(-3)) $

First, find $ g(-3) $:
$$
g(-3) = 3(-3) = -9
$$

Now plug into $ f $:
$$
f(-9) = 2(-9) - 1 = -18 - 1 = \boxed{-19}
$$

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#### 2. $ f(h(7)) $

First, $ h(7) = 7^2 + 1 = 49 + 1 = 50 $

Then $ f(50) = 2(50) - 1 = 100 - 1 = \boxed{99} $

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#### 3. $ g(h(24)) $

$ h(24) = 24^2 + 1 = 576 + 1 = 577 $

$ g(577) = 3(577) = \boxed{1731} $

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#### 4. $ h(f(9)) $

$ f(9) = 2(9) - 1 = 18 - 1 = 17 $

$ h(17) = 17^2 + 1 = 289 + 1 = \boxed{290} $

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#### 5. $ g(f(0)) $

$ f(0) = 2(0) - 1 = -1 $

$ g(-1) = 3(-1) = \boxed{-3} $

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#### 6. $ h(g(-4)) $

$ g(-4) = 3(-4) = -12 $

$ h(-12) = (-12)^2 + 1 = 144 + 1 = \boxed{145} $

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#### 7. $ f(g(h(2))) $

Start from inside:

- $ h(2) = 2^2 + 1 = 4 + 1 = 5 $
- $ g(5) = 3(5) = 15 $
- $ f(15) = 2(15) - 1 = 30 - 1 = \boxed{29} $

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#### 8. $ h(g(f(5))) $

- $ f(5) = 2(5) - 1 = 10 - 1 = 9 $
- $ g(9) = 3(9) = 27 $
- $ h(27) = 27^2 + 1 = 729 + 1 = \boxed{730} $

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#### 9. $ g(f(h(-6))) $

- $ h(-6) = (-6)^2 + 1 = 36 + 1 = 37 $
- $ f(37) = 2(37) - 1 = 74 - 1 = 73 $
- $ g(73) = 3(73) = \boxed{219} $

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Part II:

Given:
- $ f(x) = 9 - x $
- $ g(x) = x^2 + x $
- $ h(x) = x - 2 $

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#### 10. $ g(f(3)) $

- $ f(3) = 9 - 3 = 6 $
- $ g(6) = 6^2 + 6 = 36 + 6 = \boxed{42} $

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#### 11. $ f(g(4)) $

- $ g(4) = 4^2 + 4 = 16 + 4 = 20 $
- $ f(20) = 9 - 20 = \boxed{-11} $

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#### 12. $ h(f(-6)) $

- $ f(-6) = 9 - (-6) = 9 + 6 = 15 $
- $ h(15) = 15 - 2 = \boxed{13} $

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#### 13. $ f(h(-3)) $

- $ h(-3) = -3 - 2 = -5 $
- $ f(-5) = 9 - (-5) = 9 + 5 = \boxed{14} $

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#### 14. $ h(g(11)) $

- $ g(11) = 11^2 + 11 = 121 + 11 = 132 $
- $ h(132) = 132 - 2 = \boxed{130} $

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#### 15. $ g(h(-9)) $

- $ h(-9) = -9 - 2 = -11 $
- $ g(-11) = (-11)^2 + (-11) = 121 - 11 = \boxed{110} $

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#### 16. $ g(h(f(5))) $

- $ f(5) = 9 - 5 = 4 $
- $ h(4) = 4 - 2 = 2 $
- $ g(2) = 2^2 + 2 = 4 + 2 = \boxed{6} $

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#### 17. $ h(g(f(13))) $

- $ f(13) = 9 - 13 = -4 $
- $ g(-4) = (-4)^2 + (-4) = 16 - 4 = 12 $
- $ h(12) = 12 - 2 = \boxed{10} $

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#### 18. $ f(g(h(-8))) $

- $ h(-8) = -8 - 2 = -10 $
- $ g(-10) = (-10)^2 + (-10) = 100 - 10 = 90 $
- $ f(90) = 9 - 90 = \boxed{-81} $

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✔ Final Answers:

#### Part I:
1. $ f(g(-3)) = \boxed{-19} $
2. $ f(h(7)) = \boxed{99} $
3. $ g(h(24)) = \boxed{1731} $
4. $ h(f(9)) = \boxed{290} $
5. $ g(f(0)) = \boxed{-3} $
6. $ h(g(-4)) = \boxed{145} $
7. $ f(g(h(2))) = \boxed{29} $
8. $ h(g(f(5))) = \boxed{730} $
9. $ g(f(h(-6))) = \boxed{219} $

#### Part II:
10. $ g(f(3)) = \boxed{42} $
11. $ f(g(4)) = \boxed{-11} $
12. $ h(f(-6)) = \boxed{13} $
13. $ f(h(-3)) = \boxed{14} $
14. $ h(g(11)) = \boxed{130} $
15. $ g(h(-9)) = \boxed{110} $
16. $ g(h(f(5))) = \boxed{6} $
17. $ h(g(f(13))) = \boxed{10} $
18. $ f(g(h(-8))) = \boxed{-81} $

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Let me know if you'd like a visual explanation or help understanding how composition works!