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If \( f(x) = x - 1 \) and \( g(x) = 2x + 1 \), find \( f(g(x)) \) and \( f(g(2)) \).

If [tex]\( h(x) = 2x + 3 \)[/tex] and [tex]\( h^{-1}(7) = k \)[/tex], find the value of [tex]\( k \)[/tex].



Answer :

GinnyAnswer
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]

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