Question

For f left parenthesis x right parenthesis equals 8 x and g left parenthesis x right parenthesis equals x plus 5 comma find the following functions.
a. left parenthesis f circle g right parenthesis left parenthesis x right parenthesis​; b. left parenthesis g circle f right parenthesis left parenthesis x right parenthesis​; c. left parenthesis f circle g right parenthesis left parenthesis 3 right parenthesis​; d. left parenthesis g circle f right parenthesis left parenthesis 3 right parenthesis
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Part 1
a. left parenthesis f circle g right parenthesis left parenthesis x right parenthesisequals
  
enter your response here
​(Simplify your​ answer.)

Answer 1

To find the composite functions \( (f \circ g)(x) \) and \( (g \circ f)(x) \), we'll first define the functions:

- \( f(x) = 8x \)
- \( g(x) = x + 5 \)

### a. Find \( (f \circ g)(x) \)

The composition \( (f \circ g)(x) \) means we evaluate \( f \) at the result of \( g(x) \):

1. Find \( g(x) \):
\[
g(x) = x + 5
\]

2. Now plug \( g(x) \) into \( f(x) \):
\[
(f \circ g)(x) = f(g(x)) = f(x + 5)
\]

3. Evaluate \( f(x + 5) \):
\[
f(x + 5) = 8(x + 5)
\]
\[
= 8x + 40
\]

Thus,
\[
(f \circ g)(x) = 8x + 40
\]

### b. Find \( (g \circ f)(x) \)

The composition \( (g \circ f)(x) \) means we evaluate \( g \) at the result of \( f(x) \):

1. Find \( f(x) \):
\[
f(x) = 8x
\]

2. Now plug \( f(x) \) into \( g(x) \):
\[
(g \circ f)(x) = g(f(x)) = g(8x)
\]

3. Evaluate \( g(8x) \):
\[
g(8x) = 8x + 5
\]

Thus,
\[
(g \circ f)(x) = 8x + 5
\]

### c. Find \( (f \circ g)(3) \)

Using the result from part a:
\[
(f \circ g)(3) = 8(3) + 40
\]
\[
= 24 + 40 = 64
\]

### d. Find \( (g \circ f)(3) \)

Using the result from part b:
\[
(g \circ f)(3) = 8(3) + 5
\]
\[
= 24 + 5 = 29
\]

### Summary of results:
- **a.** \( (f \circ g)(x) = 8x + 40 \)
- **b.** \( (g \circ f)(x) = 8x + 5 \)
- **c.** \( (f \circ g)(3) = 64 \)
- **d.** \( (g \circ f)(3) = 29 \)