Composite Function Worksheet with problems requiring computation of function compositions.

Let's solve each problem step by step using the given functions and composite function rules.

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Given Functions (For Problems 1–9):

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

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

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

Now plug into $ f(x) $:
$$
f(-9) = 2(-9) - 1 = -18 - 1 = -19
$$

✔ Answer: $ \boxed{-19} $

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2. $ f(h(7)) $

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

Now plug into $ f(x) $:
$$
f(50) = 2(50) - 1 = 100 - 1 = 99
$$

✔ Answer: $ \boxed{99} $

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3. $ (g \circ h)(24) $

This means $ g(h(24)) $

First, compute $ h(24) $:
$$
h(24) = 24^2 + 1 = 576 + 1 = 577
$$

Now compute $ g(577) $:
$$
g(577) = 3(577) = 1731
$$

✔ Answer: $ \boxed{1731} $

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4. $ f(g(h(2))) $

Work from inside out:

Step 1: $ h(2) = 2^2 + 1 = 4 + 1 = 5 $

Step 2: $ g(5) = 3(5) = 15 $

Step 3: $ f(15) = 2(15) - 1 = 30 - 1 = 29 $

✔ Answer: $ \boxed{29} $

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

From inside out:

Step 1: $ f(5) = 2(5) - 1 = 10 - 1 = 9 $

Step 2: $ g(9) = 3(9) = 27 $

Step 3: $ h(27) = 27^2 + 1 = 729 + 1 = 730 $

✔ Answer: $ \boxed{730} $

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

From inside out:

Step 1: $ h(-6) = (-6)^2 + 1 = 36 + 1 = 37 $

Step 2: $ f(37) = 2(37) - 1 = 74 - 1 = 73 $

Step 3: $ g(73) = 3(73) = 219 $

✔ Answer: $ \boxed{219} $

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7. $ f(x + 1) $

Substitute $ x+1 $ into $ f(x) $:
$$
f(x+1) = 2(x+1) - 1 = 2x + 2 - 1 = 2x + 1
$$

✔ Answer: $ \boxed{2x + 1} $

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8. $ g(3a) $

Substitute $ 3a $ into $ g(x) $:
$$
g(3a) = 3(3a) = 9a
$$

✔ Answer: $ \boxed{9a} $

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9. $ h(x - 2) $

Substitute $ x-2 $ into $ h(x) $:
$$
h(x-2) = (x-2)^2 + 1 = x^2 - 4x + 4 + 1 = x^2 - 4x + 5
$$

✔ Answer: $ \boxed{x^2 - 4x + 5} $

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New Functions for Problems 10–11:

- $ f(x) = -3x + 7 $
- $ g(x) = 2x^2 - 8 $

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10. $ f(g(x)) $

We need to substitute $ g(x) $ into $ f(x) $:
$$
f(g(x)) = f(2x^2 - 8) = -3(2x^2 - 8) + 7 = -6x^2 + 24 + 7 = -6x^2 + 31
$$

✔ Answer: $ \boxed{-6x^2 + 31} $

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11. $ (g \circ f)(x) $

This is $ g(f(x)) $. First compute $ f(x) $, then plug into $ g $.

$ f(x) = -3x + 7 $

Now:
$$
g(f(x)) = g(-3x + 7) = 2(-3x + 7)^2 - 8
$$

First expand $ (-3x + 7)^2 $:
$$
(-3x + 7)^2 = 9x^2 - 42x + 49
$$

Now multiply by 2:
$$
2(9x^2 - 42x + 49) = 18x^2 - 84x + 98
$$

Then subtract 8:
$$
18x^2 - 84x + 98 - 8 = 18x^2 - 84x + 90
$$

✔ Answer: $ \boxed{18x^2 - 84x + 90} $

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12. If $ f(x) = 3x - 5 $ and $ g(x) = x^2 $, find $ (f \circ g)(3) $

This means $ f(g(3)) $

Step 1: $ g(3) = 3^2 = 9 $

Step 2: $ f(9) = 3(9) - 5 = 27 - 5 = 22 $

✔ Answer: $ \boxed{22} $

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13. If $ f(x) = -9x - 9 $ and $ g(x) = \sqrt{x - 9} $, find $ (f \circ g)(10) $

This means $ f(g(10)) $

Step 1: $ g(10) = \sqrt{10 - 9} = \sqrt{1} = 1 $

Step 2: $ f(1) = -9(1) - 9 = -9 - 9 = -18 $

✔ Answer: $ \boxed{-18} $

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $ -19 $ |
| 2 | $ 99 $ |
| 3 | $ 1731 $ |
| 4 | $ 29 $ |
| 5 | $ 730 $ |
| 6 | $ 219 $ |
| 7 | $ 2x + 1 $ |
| 8 | $ 9a $ |
| 9 | $ x^2 - 4x + 5 $ |
| 10 | $ -6x^2 + 31 $ |
| 11 | $ 18x^2 - 84x + 90 $ |
| 12 | $ 22 $ |
| 13 | $ -18 $ |

Make sure to circle the final answers as instructed. All work shown above should be written neatly on the worksheet.

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