Certainly! Let's solve this step-by-step:
### Step 1: Determine \(fog(x)\)
We are given two functions:
1. \( f(x) = x - 1 \)
2. \( g(x) = 2x + 1 \)
To find \( fog(x) \), which means \(f(g(x))\), we need to substitute \(g(x)\) into \(f(x)\):
[tex]\[ f(g(x)) = f(2x + 1) \][/tex]
Now, apply the function \(f\) to \(2x + 1\):
[tex]\[ f(2x + 1) = (2x + 1) - 1 \][/tex]
[tex]\[ f(2x + 1) = 2x \][/tex]
Therefore, \( fog(x) = 2x \).
### Step 2: Calculate \(fog(2)\)
Now we need to evaluate \(fog(2)\):
[tex]\[ fog(2) = 2 \times 2 \][/tex]
[tex]\[ fog(2) = 4 \][/tex]
### Step 3: Find \(k\) when \( h(x) = 2x + 3 \) and \( f^{-1}(7) = k \)
Given the function:
[tex]\[ h(x) = 2x + 3 \][/tex]
We need to find the value of \(x\) when \(h(x) = 7\):
[tex]\[ 2x + 3 = 7 \][/tex]
Subtract 3 from both sides:
[tex]\[ 2x = 4 \][/tex]
Divide by 2:
[tex]\[ x = 2 \][/tex]
Therefore, \( k = 2 \).
### Summary
- The composite function \( fog(x) = 2x \) and \( fog(2) = 4 \).
- The value of \( k \) such that \( f^{-1}(7) = k \) for the function \( h(x) = 2x + 3 \) is \( k = 2 \).
So the final results are:
[tex]\[ \text{fog}(2) = 4 \][/tex]
[tex]\[ k = 2 \][/tex]
