Question

cubic & cube root functions as inverses practice
complete this assessment to review what you’ve learned. it will not count toward your grade.
true or false? the function $f(x) = \sqrt3{(x + 9)}$ is the inverse of the function $f^{-1}(x) = x^3 - 9$
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 \( f^{-1}(f(x)) = x \).
First, compute \( f(f^{-1}(x)) \). Substitute \( f^{-1}(x)=x^3 - 9 \) into \( f \):
\( f(f^{-1}(x))=\sqrt[3]{(x^3 - 9)+9}=\sqrt[3]{x^3}=x \).

Step2: Compute the other composition

Now, compute \( f^{-1}(f(x)) \). Substitute \( f(x)=\sqrt[3]{x + 9} \) into \( f^{-1} \):
\( f^{-1}(f(x))=(\sqrt[3]{x + 9})^3-9=(x + 9)-9=x \).
Since both \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \), the functions are inverses.

Answer:

Option #1: True