Practice worksheet for composition of functions with 16 problems.

Problem: Composition of Functions

The task involves performing the indicated operations for each pair of functions \( f(x) \) and \( g(x) \). Specifically, we need to compute compositions such as \( f(g(x)) \), \( g(g(x)) \), or evaluate these compositions at specific values (e.g., \( f(g(-4)) \)).

#### General Approach:
1. Understand the composition: For \( f(g(x)) \), substitute \( g(x) \) into \( f(x) \) wherever \( x \) appears in \( f(x) \).
2. Evaluate at specific points: If asked to find \( f(g(a)) \), first compute \( g(a) \), then substitute this result into \( f(x) \).

Let's solve each problem step by step.

---

Problem 1:

\[
f(x) = 10x + 4, \quad g(x) = 6x + 4
\]
Find \( f(g(-4)) \).

#### Step 1: Compute \( g(-4) \)
\[
g(x) = 6x + 4
\]
\[
g(-4) = 6(-4) + 4 = -24 + 4 = -20
\]

#### Step 2: Compute \( f(g(-4)) = f(-20) \)
\[
f(x) = 10x + 4
\]
\[
f(-20) = 10(-20) + 4 = -200 + 4 = -196
\]

#### Final Answer:
\[
\boxed{-196}
\]

---

Problem 2:

\[
f(x) = 4x + 1, \quad g(x) = 3x + 3
\]
Find \( g(g(2)) \).

#### Step 1: Compute \( g(2) \)
\[
g(x) = 3x + 3
\]
\[
g(2) = 3(2) + 3 = 6 + 3 = 9
\]

#### Step 2: Compute \( g(g(2)) = g(9) \)
\[
g(x) = 3x + 3
\]
\[
g(9) = 3(9) + 3 = 27 + 3 = 30
\]

#### Final Answer:
\[
\boxed{30}
\]

---

Problem 3:

\[
f(x) = 5x + 4, \quad g(x) = 2x + 4
\]
Find \( f(g(-1)) \).

#### Step 1: Compute \( g(-1) \)
\[
g(x) = 2x + 4
\]
\[
g(-1) = 2(-1) + 4 = -2 + 4 = 2
\]

#### Step 2: Compute \( f(g(-1)) = f(2) \)
\[
f(x) = 5x + 4
\]
\[
f(2) = 5(2) + 4 = 10 + 4 = 14
\]

#### Final Answer:
\[
\boxed{14}
\]

---

Problem 4:

\[
f(x) = 7x + 2, \quad g(x) = 2x + 3
\]
Find \( g(g(4)) \).

#### Step 1: Compute \( g(4) \)
\[
g(x) = 2x + 3
\]
\[
g(4) = 2(4) + 3 = 8 + 3 = 11
\]

#### Step 2: Compute \( g(g(4)) = g(11) \)
\[
g(x) = 2x + 3
\]
\[
g(11) = 2(11) + 3 = 22 + 3 = 25
\]

#### Final Answer:
\[
\boxed{25}
\]

---

Problem 5:

\[
f(x) = 2x + 4, \quad g(x) = 6x + 2
\]
Find \( g(g(2)) \).

#### Step 1: Compute \( g(2) \)
\[
g(x) = 6x + 2
\]
\[
g(2) = 6(2) + 2 = 12 + 2 = 14
\]

#### Step 2: Compute \( g(g(2)) = g(14) \)
\[
g(x) = 6x + 2
\]
\[
g(14) = 6(14) + 2 = 84 + 2 = 86
\]

#### Final Answer:
\[
\boxed{86}
\]

---

Problem 6:

\[
f(x) = 3x + 6, \quad g(x) = 3x + 3
\]
Find \( f(g(-2)) \).

#### Step 1: Compute \( g(-2) \)
\[
g(x) = 3x + 3
\]
\[
g(-2) = 3(-2) + 3 = -6 + 3 = -3
\]

#### Step 2: Compute \( f(g(-2)) = f(-3) \)
\[
f(x) = 3x + 6
\]
\[
f(-3) = 3(-3) + 6 = -9 + 6 = -3
\]

#### Final Answer:
\[
\boxed{-3}
\]

---

Problem 7:

\[
f(x) = 6x + 7, \quad g(x) = 4x + 2
\]
Find \( f(g(4)) \).

#### Step 1: Compute \( g(4) \)
\[
g(x) = 4x + 2
\]
\[
g(4) = 4(4) + 2 = 16 + 2 = 18
\]

#### Step 2: Compute \( f(g(4)) = f(18) \)
\[
f(x) = 6x + 7
\]
\[
f(18) = 6(18) + 7 = 108 + 7 = 115
\]

#### Final Answer:
\[
\boxed{115}
\]

---

Problem 8:

\[
f(x) = 5x + 4, \quad g(x) = 7x + 2
\]
Find \( f(g(x)) \).

#### Step 1: Substitute \( g(x) \) into \( f(x) \)
\[
f(x) = 5x + 4, \quad g(x) = 7x + 2
\]
\[
f(g(x)) = f(7x + 2)
\]
\[
f(7x + 2) = 5(7x + 2) + 4
\]
\[
= 35x + 10 + 4
\]
\[
= 35x + 14
\]

#### Final Answer:
\[
\boxed{35x + 14}
\]

---

Problem 9:

\[
f(x) = 4x + 2, \quad g(x) = 5x + 3
\]
Find \( f(g(2)) \).

#### Step 1: Compute \( g(2) \)
\[
g(x) = 5x + 3
\]
\[
g(2) = 5(2) + 3 = 10 + 3 = 13
\]

#### Step 2: Compute \( f(g(2)) = f(13) \)
\[
f(x) = 4x + 2
\]
\[
f(13) = 4(13) + 2 = 52 + 2 = 54
\]

#### Final Answer:
\[
\boxed{54}
\]

---

Problem 10:

\[
f(x) = 11x + 1, \quad g(x) = 6x + 1
\]
Find \( g(g(1)) \).

#### Step 1: Compute \( g(1) \)
\[
g(x) = 6x + 1
\]
\[
g(1) = 6(1) + 1 = 6 + 1 = 7
\]

#### Step 2: Compute \( g(g(1)) = g(7) \)
\[
g(x) = 6x + 1
\]
\[
g(7) = 6(7) + 1 = 42 + 1 = 43
\]

#### Final Answer:
\[
\boxed{43}
\]

---

Problem 11:

\[
f(x) = 5x + 3, \quad g(x) = 6x + 4
\]
Find \( g(g(2)) \).

#### Step 1: Compute \( g(2) \)
\[
g(x) = 6x + 4
\]
\[
g(2) = 6(2) + 4 = 12 + 4 = 16
\]

#### Step 2: Compute \( g(g(2)) = g(16) \)
\[
g(x) = 6x + 4
\]
\[
g(16) = 6(16) + 4 = 96 + 4 = 100
\]

#### Final Answer:
\[
\boxed{100}
\]

---

Problem 12:

\[
f(x) = 11x + 7, \quad g(x) = 6x + 4
\]
Find \( f(g(-4)) \).

#### Step 1: Compute \( g(-4) \)
\[
g(x) = 6x + 4
\]
\[
g(-4) = 6(-4) + 4 = -24 + 4 = -20
\]

#### Step 2: Compute \( f(g(-4)) = f(-20) \)
\[
f(x) = 11x + 7
\]
\[
f(-20) = 11(-20) + 7 = -220 + 7 = -213
\]

#### Final Answer:
\[
\boxed{-213}
\]

---

Problem 13:

\[
f(x) = 8x + 8, \quad g(x) = 4x + 4
\]
Find \( g(g(4)) \).

#### Step 1: Compute \( g(4) \)
\[
g(x) = 4x + 4
\]
\[
g(4) = 4(4) + 4 = 16 + 4 = 20
\]

#### Step 2: Compute \( g(g(4)) = g(20) \)
\[
g(x) = 4x + 4
\]
\[
g(20) = 4(20) + 4 = 80 + 4 = 84
\]

#### Final Answer:
\[
\boxed{84}
\]

---

Problem 14:

\[
f(x) = 11x + 5, \quad g(x) = 8x + 2
\]
Find \( f(g(x)) \).

#### Step 1: Substitute \( g(x) \) into \( f(x) \)
\[
f(x) = 11x + 5, \quad g(x) = 8x + 2
\]
\[
f(g(x)) = f(8x + 2)
\]
\[
f(8x + 2) = 11(8x + 2) + 5
\]
\[
= 88x + 22 + 5
\]
\[
= 88x + 27
\]

#### Final Answer:
\[
\boxed{88x + 27}
\]

---

Problem 15:

\[
f(x) = 5x + 7, \quad g(x) = 7x + 3
\]
Find \( f(f(x)) \).

#### Step 1: Substitute \( f(x) \) into itself
\[
f(x) = 5x + 7
\]
\[
f(f(x)) = f(5x + 7)
\]
\[
f(5x + 7) = 5(5x + 7) + 7
\]
\[
= 25x + 35 + 7
\]
\[
= 25x + 42
\]

#### Final Answer:
\[
\boxed{25x + 42}
\]

---

Problem 16:

\[
f(x) = 10x + 7, \quad g(x) = 6x + 4
\]
Find \( g(g(3)) \).

#### Step 1: Compute \( g(3) \)
\[
g(x) = 6x + 4
\]
\[
g(3) = 6(3) + 4 = 18 + 4 = 22
\]

#### Step 2: Compute \( g(g(3)) = g(22) \)
\[
g(x) = 6x + 4
\]
\[
g(22) = 6(22) + 4 = 132 + 4 = 136
\]

#### Final Answer:
\[
\boxed{136}
\]

---

Final Answers:

\[
\boxed{
\begin{aligned}
1) & \ -196 \\
2) & \ 30 \\
3) & \ 14 \\
4) & \ 25 \\
5) & \ 86 \\
6) & \ -3 \\
7) & \ 115 \\
8) & \ 35x + 14 \\
9) & \ 54 \\
10) & \ 43 \\
11) & \ 100 \\
12) & \ -213 \\
13) & \ 84 \\
14) & \ 88x + 27 \\
15) & \ 25x + 42 \\
16) & \ 136 \\
\end{aligned}
}
\]