Sure! Let's break this problem down step-by-step.
### Step 1: Identify [tex]\( f(x) \)[/tex]
We are given two functions:
[tex]\[ g(x) = 2x + 2 \][/tex]
[tex]\[ f \circ g(x) = 8x + 13 \][/tex]
Our goal is to find the function [tex]\( f(x) \)[/tex] such that [tex]\( f(g(x)) = 8x + 13 \)[/tex].
We start by substituting [tex]\( g(x) \)[/tex] into [tex]\( f \)[/tex]:
[tex]\[ f(g(x)) = 8x + 13 \][/tex]
[tex]\[ f(2x + 2) = 8x + 13 \][/tex]
To find [tex]\( f(y) \)[/tex], let [tex]\( y = 2x + 2 \)[/tex]. So, we need to express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]:
[tex]\[ y = 2x + 2 \][/tex]
[tex]\[ y - 2 = 2x \][/tex]
[tex]\[ x = \frac{y - 2}{2} \][/tex]
Now substitute this expression for [tex]\( x \)[/tex] back into the right-hand side of [tex]\( f(g(x)) \)[/tex]:
[tex]\[ f(y) = 8 \left( \frac{y - 2}{2} \right) + 13 \][/tex]
[tex]\[ f(y) = 8 \left( \frac{y - 2}{2} \right) + 13 \][/tex]
[tex]\[ f(y) = 4(y - 2) + 13 \][/tex]
[tex]\[ f(y) = 4y - 8 + 13 \][/tex]
[tex]\[ f(y) = 4y + 5 \][/tex]
So, we have:
[tex]\[ f(x) = 8x + 9 \][/tex]
### Step 2: Set up the equation [tex]\( g(f(x)) = 28 \)[/tex]
Now we need to find the value of [tex]\( x \)[/tex] such that [tex]\( g(f(x)) = 28 \)[/tex].
Firstly, we substitute [tex]\( f(x) \)[/tex] into [tex]\( g \)[/tex]:
[tex]\[ g(f(x)) = g(8x + 9) \][/tex]
Using [tex]\( g(x) = 2x + 2 \)[/tex], we get:
[tex]\[ g(8x + 9) = 2(8x + 9) + 2 \][/tex]
[tex]\[ g(8x + 9) = 16x + 18 + 2 \][/tex]
[tex]\[ g(8x + 9) = 16x + 20 \][/tex]
We need [tex]\( g(f(x)) = 28 \)[/tex], so:
[tex]\[ 16x + 20 = 28 \][/tex]
### Step 3: Solve the equation
To find [tex]\( x \)[/tex], we solve the equation:
[tex]\[ 16x + 20 = 28 \][/tex]
[tex]\[ 16x = 28 - 20 \][/tex]
[tex]\[ 16x = 8 \][/tex]
[tex]\[ x = \frac{8}{16} \][/tex]
[tex]\[ x = \frac{1}{2} \][/tex]
### Conclusion:
The value of [tex]\( x \)[/tex] that satisfies [tex]\( g(f(x)) = 28 \)[/tex] is:
[tex]\[ x = \frac{1}{2} \][/tex]
