Question

question
for the function $f(x) = x^7 - 1$, find $f^{-1}(x)$.

answer

• $f^{-1}(x) = \sqrt7{x - 1}$
• $f^{-1}(x) = (x + 1)^7$
• $f^{-1}(x) = \sqrt7{x + 1}$
• $f^{-1}(x) = x^7 + 1$

Explanation:

Step1: Let \( y = f(x) \)

Set \( y = x^7 - 1 \).

Step2: Solve for \( x \) in terms of \( y \)

Add 1 to both sides: \( y + 1 = x^7 \).
Take the 7th root of both sides: \( x = \sqrt[7]{y + 1} \).

Step3: Replace \( x \) with \( f^{-1}(x) \) and \( y \) with \( x \)

We get \( f^{-1}(x) = \sqrt[7]{x + 1} \).

Answer:

\( f^{-1}(x) = \sqrt[7]{x + 1} \) (the third option)