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 \( x \):
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Add 6 to both sides: \[ y^2 + 6 = 2x \]
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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:
\[ f^{-1}(x) = \frac{x^2 + 6}{2}, \quad x \geq 0 \]