Let’s carefully solve for the inverse of the function $f(x) = \sqrt{6x - 8} + 5$.

Step-by-Step Solution:

To find the inverse function $f^{-1}(x)$, we follow these steps:

1. Replace $f(x)$ with $y$:
   $y = \sqrt{6x - 8} + 5$

2. Swap $x$ and $y$ to begin solving for $y$:
   $x = \sqrt{6y - 8} + 5$

3. Isolate the square root term:
   $x - 5 = \sqrt{6y - 8}$

4. Square both sides:
   $(x - 5)^2 = 6y - 8$

5. Solve for $y$:
   $6y = (x - 5)^2 + 8$ $y = \frac{(x - 5)^2 + 8}{6}$

So, the inverse function is: $f^{-1}(x) = \frac{(x - 5)^2 + 8}{6}$

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Answers in the Required Form:

• Numerator of $f^{-1}(x)$: $(x - 5)^2 + 8$
• Denominator of $f^{-1}(x)$: 6

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Would you like further breakdown of the squaring step or any other part?

5 Related Questions:

1. How do you verify if two functions are inverses of each other?
2. What are the domain and range of the inverse function?
3. How does squaring affect the original function’s domain?
4. What happens if the inverse function is not 1-to-1?
5. How do transformations like square roots and shifts affect inverses?

Tip:

Always remember to swap $x$ and $y$ when finding the inverse function. It helps visualize reversing the operations applied by the original function.