Composite function worksheet with step-by-step solutions and answers.

Problem Overview:

The worksheet involves computing composite functions and evaluating them at specific points. The tasks are divided into two sections:
1. Section 1 (Problems 1-9): Using the functions \( f(x) = 2x - 1 \), \( g(x) = 3x \), and \( h(x) = x^2 + 1 \).
2. Section 2 (Problems 10-13): Using different sets of functions.

We will solve each problem step by step.

---

Section 1: Problems 1-9

#### Given Functions:
- \( f(x) = 2x - 1 \)
- \( g(x) = 3x \)
- \( h(x) = x^2 + 1 \)

#### Problem 1: Compute \( f(g(-3)) \)

1. First, compute \( g(-3) \):
\[
g(x) = 3x \implies g(-3) = 3(-3) = -9
\]

2. Next, compute \( f(g(-3)) = f(-9) \):
\[
f(x) = 2x - 1 \implies f(-9) = 2(-9) - 1 = -18 - 1 = -19
\]

Answer:
\[
\boxed{-19}
\]

#### Problem 2: Compute \( f(h(7)) \)

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

2. Next, compute \( f(h(7)) = f(50) \):
\[
f(x) = 2x - 1 \implies f(50) = 2(50) - 1 = 100 - 1 = 99
\]

Answer:
\[
\boxed{99}
\]

#### Problem 3: Compute \( (g \circ h)(24) \)

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

2. Next, compute \( g(h(24)) = g(577) \):
\[
g(x) = 3x \implies g(577) = 3(577) = 1731
\]

Answer:
\[
\boxed{1731}
\]

#### Problem 4: Compute \( f(g(h(2))) \)

1. First, compute \( h(2) \):
\[
h(x) = x^2 + 1 \implies h(2) = 2^2 + 1 = 4 + 1 = 5
\]

2. Next, compute \( g(h(2)) = g(5) \):
\[
g(x) = 3x \implies g(5) = 3(5) = 15
\]

3. Finally, compute \( f(g(h(2))) = f(15) \):
\[
f(x) = 2x - 1 \implies f(15) = 2(15) - 1 = 30 - 1 = 29
\]

Answer:
\[
\boxed{29}
\]

#### Problem 5: Compute \( h(g(f(5))) \)

1. First, compute \( f(5) \):
\[
f(x) = 2x - 1 \implies f(5) = 2(5) - 1 = 10 - 1 = 9
\]

2. Next, compute \( g(f(5)) = g(9) \):
\[
g(x) = 3x \implies g(9) = 3(9) = 27
\]

3. Finally, compute \( h(g(f(5))) = h(27) \):
\[
h(x) = x^2 + 1 \implies h(27) = 27^2 + 1 = 729 + 1 = 730
\]

Answer:
\[
\boxed{730}
\]

#### Problem 6: Compute \( g(f(h(-6))) \)

1. First, compute \( h(-6) \):
\[
h(x) = x^2 + 1 \implies h(-6) = (-6)^2 + 1 = 36 + 1 = 37
\]

2. Next, compute \( f(h(-6)) = f(37) \):
\[
f(x) = 2x - 1 \implies f(37) = 2(37) - 1 = 74 - 1 = 73
\]

3. Finally, compute \( g(f(h(-6))) = g(73) \):
\[
g(x) = 3x \implies g(73) = 3(73) = 219
\]

Answer:
\[
\boxed{219}
\]

#### Problem 7: Compute \( f(x+1) \)

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

Answer:
\[
\boxed{2x + 1}
\]

#### Problem 8: Compute \( g(3a) \)

1. Substitute \( 3a \) into \( g(x) \):
\[
g(x) = 3x \implies g(3a) = 3(3a) = 9a
\]

Answer:
\[
\boxed{9a}
\]

#### Problem 9: Compute \( h(x-2) \)

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

2. Expand \( (x-2)^2 \):
\[
(x-2)^2 = x^2 - 4x + 4
\]

3. Add 1:
\[
h(x-2) = x^2 - 4x + 4 + 1 = x^2 - 4x + 5
\]

Answer:
\[
\boxed{x^2 - 4x + 5}
\]

---

Section 2: Problems 10-13

#### Problem 10: Compute \( f(g(x)) \) for \( f(x) = -3x + 7 \) and \( g(x) = 2x^2 - 8 \)

1. Substitute \( g(x) \) into \( f(x) \):
\[
f(x) = -3x + 7 \quad \text{and} \quad g(x) = 2x^2 - 8
\]
\[
f(g(x)) = f(2x^2 - 8) = -3(2x^2 - 8) + 7
\]

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

Answer:
\[
\boxed{-6x^2 + 31}
\]

#### Problem 11: Compute \( (g \circ f)(x) \) for \( f(x) = -3x + 7 \) and \( g(x) = 2x^2 - 8 \)

1. Substitute \( f(x) \) into \( g(x) \):
\[
f(x) = -3x + 7 \quad \text{and} \quad g(x) = 2x^2 - 8
\]
\[
(g \circ f)(x) = g(f(x)) = g(-3x + 7)
\]

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

3. Expand \( (-3x + 7)^2 \):
\[
(-3x + 7)^2 = (-3x)^2 + 2(-3x)(7) + 7^2 = 9x^2 - 42x + 49
\]

4. Substitute back:
\[
g(-3x + 7) = 2(9x^2 - 42x + 49) - 8 = 18x^2 - 84x + 98 - 8 = 18x^2 - 84x + 90
\]

Answer:
\[
\boxed{18x^2 - 84x + 90}
\]

#### Problem 12: Compute \( (f \circ g)(3) \) for \( f(x) = 3x - 5 \) and \( g(x) = x^2 \)

1. First, compute \( g(3) \):
\[
g(x) = x^2 \implies g(3) = 3^2 = 9
\]

2. Next, compute \( f(g(3)) = f(9) \):
\[
f(x) = 3x - 5 \implies f(9) = 3(9) - 5 = 27 - 5 = 22
\]

Answer:
\[
\boxed{22}
\]

#### Problem 13: Compute \( (f \circ g)(10) \) for \( f(x) = -9x - 9 \) and \( g(x) = \sqrt{x-9} \)

1. First, compute \( g(10) \):
\[
g(x) = \sqrt{x-9} \implies g(10) = \sqrt{10-9} = \sqrt{1} = 1
\]

2. Next, compute \( f(g(10)) = f(1) \):
\[
f(x) = -9x - 9 \implies f(1) = -9(1) - 9 = -9 - 9 = -18
\]

Answer:
\[
\boxed{-18}
\]

---

Final Answers:

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