Composite Function Worksheet PDF Form - FormsPal - Free Printable

Problem: Composite Function Worksheet

We are given three functions:
- \( f(x) = 2x - 1 \)
- \( g(x) = 3x \)
- \( h(x) = x^2 + 1 \)

We need to compute the following composite functions and expressions:

1. \( f(g(-3)) \)
2. \( f(h(7)) \)
3. \( (g \circ h)(24) \)
4. \( f(g(h(2))) \)
5. \( h(g(f(5))) \)
6. \( g(f(h(-6))) \)
7. \( f(x + 1) \)
8. \( g(3a) \)
9. \( h(x - 2) \)

Solution:

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

#### 2. \( f(h(7)) \)
First, compute \( h(7) \):
\[
h(x) = x^2 + 1 \implies h(7) = 7^2 + 1 = 49 + 1 = 50
\]
Next, compute \( f(h(7)) = f(50) \):
\[
f(x) = 2x - 1 \implies f(50) = 2(50) - 1 = 100 - 1 = 99
\]
Thus, \( f(h(7)) = 99 \).

#### 3. \( (g \circ h)(24) \)
This is equivalent to \( g(h(24)) \). First, compute \( h(24) \):
\[
h(x) = x^2 + 1 \implies h(24) = 24^2 + 1 = 576 + 1 = 577
\]
Next, compute \( g(h(24)) = g(577) \):
\[
g(x) = 3x \implies g(577) = 3(577) = 1731
\]
Thus, \( (g \circ h)(24) = 1731 \).

#### 4. \( f(g(h(2))) \)
First, compute \( h(2) \):
\[
h(x) = x^2 + 1 \implies h(2) = 2^2 + 1 = 4 + 1 = 5
\]
Next, compute \( g(h(2)) = g(5) \):
\[
g(x) = 3x \implies g(5) = 3(5) = 15
\]
Finally, compute \( f(g(h(2))) = f(15) \):
\[
f(x) = 2x - 1 \implies f(15) = 2(15) - 1 = 30 - 1 = 29
\]
Thus, \( f(g(h(2))) = 29 \).

#### 5. \( h(g(f(5))) \)
First, compute \( f(5) \):
\[
f(x) = 2x - 1 \implies f(5) = 2(5) - 1 = 10 - 1 = 9
\]
Next, compute \( g(f(5)) = g(9) \):
\[
g(x) = 3x \implies g(9) = 3(9) = 27
\]
Finally, compute \( h(g(f(5))) = h(27) \):
\[
h(x) = x^2 + 1 \implies h(27) = 27^2 + 1 = 729 + 1 = 730
\]
Thus, \( h(g(f(5))) = 730 \).

#### 6. \( g(f(h(-6))) \)
First, compute \( h(-6) \):
\[
h(x) = x^2 + 1 \implies h(-6) = (-6)^2 + 1 = 36 + 1 = 37
\]
Next, compute \( f(h(-6)) = f(37) \):
\[
f(x) = 2x - 1 \implies f(37) = 2(37) - 1 = 74 - 1 = 73
\]
Finally, compute \( g(f(h(-6))) = g(73) \):
\[
g(x) = 3x \implies g(73) = 3(73) = 219
\]
Thus, \( g(f(h(-6))) = 219 \).

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

#### 8. \( g(3a) \)
Substitute \( 3a \) into \( g(x) \):
\[
g(x) = 3x \implies g(3a) = 3(3a) = 9a
\]
Thus, \( g(3a) = 9a \).

#### 9. \( h(x - 2) \)
Substitute \( x - 2 \) into \( h(x) \):
\[
h(x) = x^2 + 1 \implies h(x - 2) = (x - 2)^2 + 1 = x^2 - 4x + 4 + 1 = x^2 - 4x + 5
\]
Thus, \( h(x - 2) = x^2 - 4x + 5 \).

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \ -19 \\
2. & \ 99 \\
3. & \ 1731 \\
4. & \ 29 \\
5. & \ 730 \\
6. & \ 219 \\
7. & \ 2x + 1 \\
8. & \ 9a \\
9. & \ x^2 - 4x + 5
\end{aligned}
}
\]