Question

cubic & cube root functions as inverses practice
complete this assessment to review what youve learned. it will not count toward your
the function $f(x) = x^3 - 1$ is the inverse of the function $f^{-1}(x) = \sqrt3{(x + 1)}$.
option #1: true
option #2: false
(1 point)
the best answer is option #\square.
check answer remaining attempts : 3

Explanation:

Step1: Recall inverse function composition

To check if \( f(x) \) and \( f^{-1}(x) \) are inverses, we verify \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

Step2: Compute \( f(f^{-1}(x)) \)

Substitute \( f^{-1}(x) = \sqrt[3]{x + 1} \) into \( f(x) \):
\( f(f^{-1}(x)) = (\sqrt[3]{x + 1})^3 - 1 \).
Simplify: \( (x + 1) - 1 = x \).

Step3: Compute \( f^{-1}(f(x)) \)

Substitute \( f(x) = x^3 - 1 \) into \( f^{-1}(x) \):
\( f^{-1}(f(x)) = \sqrt[3]{(x^3 - 1) + 1} \).
Simplify: \( \sqrt[3]{x^3} = x \).

Since both compositions equal \( x \), \( f(x) \) and \( f^{-1}(x) \) are inverses.

Answer:

Option #1: True