Question

Find ​(fcircle​g)(x) and ​(gcircle​f)(x).
​f(x)equals5xminus8​, ​ g(x)equals4minus4x
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Part 1
​(fcircle​g)(x)equals
  
enter your response here ​(Simplify your​ answer.)
Part 2
​(gcircle​f)(x)equals
  
enter your response here ​(Simplify your​ answer.)

Answer 1

To find \((f \circ g)(x)\) and \((g \circ f)(x)\), we first need to understand how function composition works. The notation \((f \circ g)(x)\) means that we will evaluate \(g(x)\) and then plug that result into \(f\). Similarly, \((g \circ f)(x)\) means that we will evaluate \(f(x)\) and then plug that result into \(g\).

Given:
- \(f(x) = 5x - 8\)
- \(g(x) = 4 - 4x\)

### Part 1: Calculate \((f \circ g)(x)\)
1. First, find \(g(x)\):
\[
g(x) = 4 - 4x
\]

2. Now, substitute \(g(x)\) into \(f\):
\[
(f \circ g)(x) = f(g(x)) = f(4 - 4x)
\]

3. Now apply the function \(f\):
\[
f(4 - 4x) = 5(4 - 4x) - 8
\]
\[
= 20 - 20x - 8
\]
\[
= 12 - 20x
\]

So, \((f \circ g)(x) = 12 - 20x\).

### Part 2: Calculate \((g \circ f)(x)\)
1. First, find \(f(x)\):
\[
f(x) = 5x - 8
\]

2. Now, substitute \(f(x)\) into \(g\):
\[
(g \circ f)(x) = g(f(x)) = g(5x - 8)
\]

3. Now apply the function \(g\):
\[
g(5x - 8) = 4 - 4(5x - 8)
\]
\[
= 4 - 20x + 32
\]
\[
= 36 - 20x
\]

So, \((g \circ f)(x) = 36 - 20x\).

### Summary of the results:
- \((f \circ g)(x) = 12 - 20x\)
- \((g \circ f)(x) = 36 - 20x\)

You can enter these results in the respective response fields as:
- \((f \circ g)(x) = 12 - 20x\)
- \((g \circ f)(x) = 36 - 20x\)