Question

For f left parenthesis x right parenthesis equals 5 x and g left parenthesis x right parenthesis equals x plus 2 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 4 right parenthesis​; d. left parenthesis g circle f right parenthesis left parenthesis 4 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.)
Part 2
b. left parenthesis g circle f right parenthesis left parenthesis x right parenthesisequals
  
enter your response here
​(Simplify your​ answer.)
Part 3
c. left parenthesis f circle g right parenthesis left parenthesis 4 right parenthesisequals
  
enter your response here
Part 4
d. left parenthesis g circle f right parenthesis left parenthesis 4 right parenthesisequals
  
enter your response here

Answer 1

To find the composite functions and their specific values, let's go step-by-step through the calculations.

### Given functions:
- \( f(x) = 5x \)
- \( g(x) = x + 2 \)

### Part 1: \( (f \circ g)(x) \)

This means we will apply the function \( g \) first, and then apply the function \( f \) to the result.

\[
(f \circ g)(x) = f(g(x)) = f(x + 2)
\]

Substituting \( g(x) \) into \( f \):

\[
f(g(x)) = f(x + 2) = 5(x + 2) = 5x + 10
\]

Thus,

\[
(f \circ g)(x) = 5x + 10
\]

### Part 2: \( (g \circ f)(x) \)

This means we will apply the function \( f \) first, and then apply the function \( g \) to the result.

\[
(g \circ f)(x) = g(f(x)) = g(5x)
\]

Substituting \( f(x) \) into \( g \):

\[
g(f(x)) = g(5x) = 5x + 2
\]

Thus,

\[
(g \circ f)(x) = 5x + 2
\]

### Part 3: \( (f \circ g)(4) \)

Now we will evaluate \( (f \circ g)(x) \) at \( x = 4 \):

\[
(f \circ g)(4) = 5(4) + 10 = 20 + 10 = 30
\]

### Part 4: \( (g \circ f)(4) \)

Now we will evaluate \( (g \circ f)(x) \) at \( x = 4 \):

\[
(g \circ f)(4) = 5(4) + 2 = 20 + 2 = 22
\]

### Summary of the Results:
a. \( (f \circ g)(x) = 5x + 10 \)

b. \( (g \circ f)(x) = 5x + 2 \)

c. \( (f \circ g)(4) = 30 \)

d. \( (g \circ f)(4) = 22 \)