Question

Let ​f(x)equalsStartFraction 8 x minus 9 Over 4 EndFraction
and ​g(x)equalsStartFraction 4 x plus 9 Over 8 EndFraction
.
​(a) Find left parenthesis f circle g right parenthesis left parenthesis x right parenthesis.
​(b) Find left parenthesis g circle f right parenthesis left parenthesis x 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.)

Answer 1

To find the compositions of the functions \( f \) and \( g \), we will handle each part separately.

1. **Function Definitions**:
- \( f(x) = \frac{8x - 9}{4} \)
- \( g(x) = \frac{4x + 9}{8} \)

---

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

To find \( (f \circ g)(x) \), we need to evaluate \( f(g(x)) \).

1. Substitute \( g(x) \) into \( f(x) \):
\[
g(x) = \frac{4x + 9}{8}
\]
So, we have:
\[
f(g(x)) = f\left(\frac{4x + 9}{8}\right)
\]

2. Now plug \( \frac{4x + 9}{8} \) into the formula for \( f(x) \):
\[
f\left(\frac{4x + 9}{8}\right) = \frac{8\left(\frac{4x + 9}{8}\right) - 9}{4}
\]

3. Simplifying this, we first compute \( 8\left(\frac{4x + 9}{8}\right) \):
\[
8\left(\frac{4x + 9}{8}\right) = 4x + 9
\]

4. Then, substituting back in, we have:
\[
f\left(\frac{4x + 9}{8}\right) = \frac{4x + 9 - 9}{4} = \frac{4x}{4} = x
\]

So, we find:
\[
(f \circ g)(x) = x
\]

---

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

To find \( (g \circ f)(x) \), we need to evaluate \( g(f(x)) \).

1. Substitute \( f(x) \) into \( g(x) \):
\[
f(x) = \frac{8x - 9}{4}
\]
So, we have:
\[
g(f(x)) = g\left(\frac{8x - 9}{4}\right)
\]

2. Plug \( \frac{8x - 9}{4} \) into the formula for \( g(x) \):
\[
g\left(\frac{8x - 9}{4}\right) = \frac{4\left(\frac{8x - 9}{4}\right) + 9}{8}
\]

3. Simplifying this, we first compute \( 4\left(\frac{8x - 9}{4}\right) \):
\[
4\left(\frac{8x - 9}{4}\right) = 8x - 9
\]

4. Then, substituting back in, we have:
\[
g\left(\frac{8x - 9}{4}\right) = \frac{8x - 9 + 9}{8} = \frac{8x}{8} = x
\]

So, we find:
\[
(g \circ f)(x) = x
\]

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### Final Answers:
- **Part 1**: \( (f \circ g)(x) = x \)
- **Part 2**: \( (g \circ f)(x) = x \)