Composite Function Worksheet - Fill Online, Printable, Fillable ... - Free Printable

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## 📌 Given Functions (for Problems 1–9):
- \( f(x) = 2x - 1 \)
- \( g(x) = 3x \)
- \( h(x) = x^2 + 4 \)

We are to compute various compositions and evaluations. Let’s go one by one.

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✔ Problem 1: \( f(g(3)) \)

First, find \( g(3) \):
> \( g(x) = 3x \Rightarrow g(3) = 3 \cdot 3 = 9 \)

Now plug into \( f \):
> \( f(9) = 2(9) - 1 = 18 - 1 = 17 \)

✔ Answer: 17

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✔ Problem 2: \( f(g(h(2))) \)

This is a triple composition: \( f(g(h(2))) \)

Step 1: Compute \( h(2) \)
> \( h(x) = x^2 + 4 \Rightarrow h(2) = 2^2 + 4 = 4 + 4 = 8 \)

Step 2: Compute \( g(h(2)) = g(8) \)
> \( g(8) = 3 \cdot 8 = 24 \)

Step 3: Compute \( f(g(h(2))) = f(24) \)
> \( f(24) = 2(24) - 1 = 48 - 1 = 47 \)

✔ Answer: 47

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✔ Problem 3: \( (g \circ h)(5) \)

This means \( g(h(5)) \)

Step 1: \( h(5) = 5^2 + 4 = 25 + 4 = 29 \)

Step 2: \( g(29) = 3 \cdot 29 = 87 \)

✔ Answer: 87

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✔ Problem 4: \( g(h(2)) \)

We already computed this in Problem 2:
> \( h(2) = 8 \), then \( g(8) = 24 \)

✔ Answer: 24

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✔ Problem 5: \( h(g(f(2))) \)

Triple composition again: \( h(g(f(2))) \)

Step 1: \( f(2) = 2(2) - 1 = 4 - 1 = 3 \)

Step 2: \( g(3) = 3 \cdot 3 = 9 \)

Step 3: \( h(9) = 9^2 + 4 = 81 + 4 = 85 \)

✔ Answer: 85

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✔ Problem 6: \( g(f(-4)) \)

Step 1: \( f(-4) = 2(-4) - 1 = -8 - 1 = -9 \)

Step 2: \( g(-9) = 3 \cdot (-9) = -27 \)

✔ Answer: -27

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✔ Problem 7: \( f(x + 1) \)

Substitute \( x+1 \) into \( f \):

> \( f(x+1) = 2(x+1) - 1 = 2x + 2 - 1 = 2x + 1 \)

✔ Answer: \( 2x + 1 \)

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✔ Problem 8: \( g(3x) \)

Substitute \( 3x \) into \( g \):

> \( g(3x) = 3 \cdot (3x) = 9x \)

✔ Answer: \( 9x \)

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✔ Problem 9: \( h(x - 2) \)

Substitute \( x-2 \) into \( h \):

> \( h(x - 2) = (x - 2)^2 + 4 = x^2 - 4x + 4 + 4 = x^2 - 4x + 8 \)

✔ Answer: \( x^2 - 4x + 8 \)

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## 🔁 Problems 10–13: New Functions

> Let \( f(x) = -3x + 7 \) and \( g(x) = 2x^2 - 8 \)

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✔ Problem 10: \( f(g(x)) \)

Substitute \( g(x) \) into \( f \):

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

Distribute:

> \( = -6x^2 + 24 + 7 = -6x^2 + 31 \)

✔ Answer: \( -6x^2 + 31 \)

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✔ Problem 11: \( (g \circ f)(x) \)

This is \( g(f(x)) \)

> \( f(x) = -3x + 7 \)

So,

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

First, square the binomial:

> \( (-3x + 7)^2 = 9x^2 - 42x + 49 \)

Multiply by 2:

> \( 2(9x^2 - 42x + 49) = 18x^2 - 84x + 98 \)

Subtract 8:

> \( 18x^2 - 84x + 98 - 8 = 18x^2 - 84x + 90 \)

✔ Answer: \( 18x^2 - 84x + 90 \)

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## ⚠️ Problems 12 & 13: Different Functions Again

> If \( f(x) = 3x - 5 \) and \( g(x) = x^3 \)

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✔ Problem 12: Find \( (f \circ g)(x) \)

This is \( f(g(x)) \)

> \( g(x) = x^3 \), so \( f(x^3) = 3(x^3) - 5 = 3x^3 - 5 \)

✔ Answer: \( 3x^3 - 5 \)

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> If \( f(x) = -9x - 9 \) and \( g(x) = \sqrt{x - 9} \)

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✔ Problem 13: Find \( (f \circ g)(10) \)

This is \( f(g(10)) \)

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

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

✔ Answer: -18

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## ✔ Final Answers Summary:

| Problem | Answer |
|---------|---------------------|
| 1 | 17 |
| 2 | 47 |
| 3 | 87 |
| 4 | 24 |
| 5 | 85 |
| 6 | -27 |
| 7 | \( 2x + 1 \) |
| 8 | \( 9x \) |
| 9 | \( x^2 - 4x + 8 \) |
| 10 | \( -6x^2 + 31 \) |
| 11 | \( 18x^2 - 84x + 90 \) |
| 12 | \( 3x^3 - 5 \) |
| 13 | -18 |

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