Question

Note: Enter your answer and show all the steps that you use to solve this problem in the space provided.

Find the inverse function for f(x)=2x−6−−−−−√
.

Answer 1

To find the inverse function of \( f(x) = \sqrt{2x - 6} \), we need to follow several steps. Here’s how to do it:

### Step 1: Rewrite the function
We start by rewriting \( f(x) \) as \( y \).

\[
y = \sqrt{2x - 6}
\]

### Step 2: Solve for \( x \)
Now, we need to solve the equation for \( x \) in terms of \( y \). To do this, first we square both sides to eliminate the square root:

\[
y^2 = 2x - 6
\]

Next, we can isolate \( x \):

1. Add 6 to both sides:
\[
y^2 + 6 = 2x
\]

2. Divide by 2:
\[
x = \frac{y^2 + 6}{2}
\]

### Step 3: Express the inverse function
Now, we express \( x \) in terms of \( y \) to denote the inverse function. By switching \( x \) and \( y \):

\[
f^{-1}(x) = \frac{x^2 + 6}{2}
\]

### Step 4: State the final result
The inverse function is:

\[
f^{-1}(x) = \frac{x^2 + 6}{2}
\]

### Step 5: Determine the domain of the inverse
Since the original function \( f(x) = \sqrt{2x - 6} \) requires that \( 2x - 6 \geq 0 \), we find the domain of \( f \):

\[
2x \geq 6 \Rightarrow x \geq 3
\]

This means the range of \( f \), which becomes the domain of \( f^{-1} \), is \( [0, \infty) \) because the square root function outputs non-negative values.

### Conclusion
Thus, the complete answer with the domain of the inverse function is:

\[
f^{-1}(x) = \frac{x^2 + 6}{2}, \quad x \geq 0
\]