Question

For f left parenthesis x right parenthesis equals 2 x minus 5 and g left parenthesis x right parenthesis equals StartFraction x plus 5 Over 2 EndFraction
​, 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 7 right parenthesis​; d. left parenthesis g circle f right parenthesis left parenthesis 7 right parenthesis
Question content area bottom
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 7 right parenthesisequals
  
enter your response here
Part 4
d. left parenthesis g circle f right parenthesis left parenthesis 7 right parenthesisequals
  
enter your response here

Answer 1

Let's work through the problems step by step.

Given:
- \( f(x) = 2x - 5 \)
- \( g(x) = \frac{x + 5}{2} \)

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

The composition \( (f \circ g)(x) \) is defined as \( f(g(x)) \).

1. Substitute \( g(x) \) into \( f(x) \):
\[
g(x) = \frac{x + 5}{2}
\]
2. Now we substitute this into \( f(x) \):
\[
f(g(x)) = f\left(\frac{x + 5}{2}\right) = 2\left(\frac{x + 5}{2}\right) - 5
\]
3. Simplify:
\[
= (x + 5) - 5 = x
\]

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

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

Now, let's calculate \( (g \circ f)(x) \) which is \( g(f(x)) \).

1. Substitute \( f(x) \) into \( g(x) \):
\[
f(x) = 2x - 5
\]
2. Now we substitute this into \( g(x) \):
\[
g(f(x)) = g(2x - 5) = \frac{(2x - 5) + 5}{2}
\]
3. Simplify:
\[
= \frac{2x}{2} = x
\]

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

### Part 3: \( (f \circ g)(7) \)

We already found that \( (f \circ g)(x) = x \), so:
\[
(f \circ g)(7) = 7
\]

### Part 4: \( (g \circ f)(7) \)

Similarly, since \( (g \circ f)(x) = x \):
\[
(g \circ f)(7) = 7
\]

### Summary of Results:
- Part 1: \( (f \circ g)(x) = x \)
- Part 2: \( (g \circ f)(x) = x \)
- Part 3: \( (f \circ g)(7) = 7 \)
- Part 4: \( (g \circ f)(7) = 7 \)