Question

For f left parenthesis x right parenthesis equals 6 x minus 4 and g left parenthesis x right parenthesis equals StartFraction x plus 4 Over 6 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 8 right parenthesis​; d. left parenthesis g circle f right parenthesis left parenthesis 8 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.)

Answer 1

To find the functions \( (f \circ g)(x) \) and \( (g \circ f)(x) \), we'll apply the definitions of functions and compositions.

Given:
\[
f(x) = 6x - 4
\]
\[
g(x) = \frac{x + 4}{6}
\]

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

This means we need to find \( f(g(x)) \):
\[
g(x) = \frac{x + 4}{6}
\]
Now, substitute \( g(x) \) into \( f(x) \):
\[
f(g(x)) = f\left(\frac{x + 4}{6}\right)
\]
Plugging this into \( f(x) \):
\[
= 6\left(\frac{x + 4}{6}\right) - 4
\]
\[
= (x + 4) - 4
\]
\[
= x
\]

So, \( (f \circ g)(x) = x \).

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

Now we need to find \( g(f(x)) \):
\[
f(x) = 6x - 4
\]
Now substitute \( f(x) \) into \( g(x) \):
\[
g(f(x)) = g(6x - 4)
\]
This gives:
\[
= \frac{(6x - 4) + 4}{6}
\]
\[
= \frac{6x}{6}
\]
\[
= x
\]

So, \( (g \circ f)(x) = x \).

### Part c: \( (f \circ g)(8) \)

The result from part a tells us:
\[
(f \circ g)(x) = x
\]
Thus,
\[
(f \circ g)(8) = 8
\]

### Part d: \( (g \circ f)(8) \)

The result from part b tells us:
\[
(g \circ f)(x) = x
\]
Thus,
\[
(g \circ f)(8) = 8
\]

### Summary of Results:
a. \( (f \circ g)(x) = x \)
b. \( (g \circ f)(x) = x \)
c. \( (f \circ g)(8) = 8 \)
d. \( (g \circ f)(8) = 8 \)