Question

for the function $f(x) = 8x^5 - 7$, find $f^{-1}(x)$.
answer
\\( f^{-1}(x) = \sqrt5{\frac{x}{8}} + 7 \\) \\( f^{-1}(x) = \frac{\sqrt5{x}}{8} + 7 \\)
\\( f^{-1}(x) = \sqrt5{\frac{x}{8} + 7} \\) \\( f^{-1}(x) = \sqrt5{\frac{x + 7}{8}} \\)

Explanation:

Step1: Replace \( f(x) \) with \( y \)

We start with the function \( f(x) = 8x^5 - 7 \). Replace \( f(x) \) with \( y \), so we have \( y = 8x^5 - 7 \).

Step2: Swap \( x \) and \( y \)

To find the inverse function, we swap the roles of \( x \) and \( y \). This gives us \( x = 8y^5 - 7 \).

Step3: Solve for \( y \)

First, add 7 to both sides of the equation: \( x + 7 = 8y^5 \). Then, divide both sides by 8: \( \frac{x + 7}{8} = y^5 \). Finally, take the fifth root of both sides to solve for \( y \): \( y = \sqrt[5]{\frac{x + 7}{8}} \).

Step4: Replace \( y \) with \( f^{-1}(x) \)

Now, replace \( y \) with \( f^{-1}(x) \) to get the inverse function: \( f^{-1}(x) = \sqrt[5]{\frac{x + 7}{8}} \).

Answer:

\( f^{-1}(x) = \sqrt[5]{\frac{x + 7}{8}} \) (the fourth option)