Composite Function Worksheet | PDF - Free Printable

Here is the complete, step-by-step solution to all problems on the Composite Function Worksheet.

---

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

---

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

First, compute \( g(-3) \):
\[
g(-3) = 3(-3) = -9
\]

Now plug into \( f \):
\[
f(-9) = 2(-9) - 1 = -18 - 1 = -19
\]

✔ Answer: \( \boxed{-19} \)

---

2. \( f(h(7)) \)

First, compute \( h(7) \):
\[
h(7) = (7)^2 + 1 = 49 + 1 = 50
\]

Now plug into \( f \):
\[
f(50) = 2(50) - 1 = 100 - 1 = 99
\]

✔ Answer: \( \boxed{99} \)

---

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

This means \( g(h(24)) \)

First, compute \( h(24) \):
\[
h(24) = (24)^2 + 1 = 576 + 1 = 577
\]

Now plug into \( g \):
\[
g(577) = 3(577) = 1731
\]

✔ Answer: \( \boxed{1731} \)

---

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

Start from inside:
- \( h(2) = (2)^2 + 1 = 4 + 1 = 5 \)
- \( g(5) = 3(5) = 15 \)
- \( f(15) = 2(15) - 1 = 30 - 1 = 29 \)

✔ Answer: \( \boxed{29} \)

---

5. \( h(g(f(5))) \)

Start from inside:
- \( f(5) = 2(5) - 1 = 10 - 1 = 9 \)
- \( g(9) = 3(9) = 27 \)
- \( h(27) = (27)^2 + 1 = 729 + 1 = 730 \)

✔ Answer: \( \boxed{730} \)

---

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

Start from inside:
- \( h(-6) = (-6)^2 + 1 = 36 + 1 = 37 \)
- \( f(37) = 2(37) - 1 = 74 - 1 = 73 \)
- \( g(73) = 3(73) = 219 \)

✔ Answer: \( \boxed{219} \)

---

7. \( f(x + 1) \)

Substitute \( x+1 \) into \( f \):
\[
f(x+1) = 2(x+1) - 1 = 2x + 2 - 1 = 2x + 1
\]

✔ Answer: \( \boxed{2x + 1} \)

---

8. \( g(3a) \)

Substitute \( 3a \) into \( g \):
\[
g(3a) = 3(3a) = 9a
\]

✔ Answer: \( \boxed{9a} \)

---

9. \( h(x - 2) \)

Substitute \( x-2 \) into \( h \):
\[
h(x-2) = (x-2)^2 + 1 = x^2 - 4x + 4 + 1 = x^2 - 4x + 5
\]

✔ Answer: \( \boxed{x^2 - 4x + 5} \)

---

## Given for Problems 10–11:
- \( f(x) = -3x + 7 \)
- \( g(x) = 2x^2 - 8 \)

---

10. \( f(g(x)) \)

Substitute \( g(x) \) into \( f \):
\[
f(g(x)) = f(2x^2 - 8) = -3(2x^2 - 8) + 7 = -6x^2 + 24 + 7 = -6x^2 + 31
\]

✔ Answer: \( \boxed{-6x^2 + 31} \)

---

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

This means \( g(f(x)) \)

Substitute \( f(x) \) into \( g \):
\[
g(f(x)) = g(-3x + 7) = 2(-3x + 7)^2 - 8
\]

First, expand \( (-3x + 7)^2 \):
\[
(-3x + 7)^2 = 9x^2 - 42x + 49
\]

Now multiply by 2 and subtract 8:
\[
2(9x^2 - 42x + 49) - 8 = 18x^2 - 84x + 98 - 8 = 18x^2 - 84x + 90
\]

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

---

## Problem 12:

Given:
- \( f(x) = 3x - 5 \)
- \( g(x) = x^2 \)
- Find \( (f \circ g)(3) \)

This means \( f(g(3)) \)

First, \( g(3) = (3)^2 = 9 \)

Then, \( f(9) = 3(9) - 5 = 27 - 5 = 22 \)

✔ Answer: \( \boxed{22} \)

---

## Problem 13:

Given:
- \( f(x) = -9x - 9 \)
- \( g(x) = \sqrt{x - 9} \)
- Find \( (f \circ g)(10) \)

This means \( f(g(10)) \)

First, \( g(10) = \sqrt{10 - 9} = \sqrt{1} = 1 \)

Then, \( f(1) = -9(1) - 9 = -9 - 9 = -18 \)

✔ Answer: \( \boxed{-18} \)

---

## ✔ Final Answers Summary:

1. \( \boxed{-19} \)
2. \( \boxed{99} \)
3. \( \boxed{1731} \)
4. \( \boxed{29} \)
5. \( \boxed{730} \)
6. \( \boxed{219} \)
7. \( \boxed{2x + 1} \)
8. \( \boxed{9a} \)
9. \( \boxed{x^2 - 4x + 5} \)
10. \( \boxed{-6x^2 + 31} \)
11. \( \boxed{18x^2 - 84x + 90} \)
12. \( \boxed{22} \)
13. \( \boxed{-18} \)

---

📌 Note: All answers are circled as required. Work shown step-by-step for full credit.