Question

1. is the inverse a function?

$f(x) = (x - 6)^3$
yes
no

1. write the inverse function in the form $a cdot sqrt3{bx + c} + d$, where $a, b, c$, and $d$ are constants. simplify any fractions.

$f^{-1}(x) = \square$

Explanation:

Step1: Check if function is one-to-one

A cubic function \(f(x)=(x-6)^3\) is strictly increasing, so it passes the horizontal line test (each \(y\) has exactly one \(x\)).

Step2: Swap \(x\) and \(y\) to find inverse

Let \(y=(x-6)^3\). Swap variables: \(x=(y-6)^3\)

Step3: Solve for \(y\)

Take cube root of both sides: \(\sqrt[3]{x}=y-6\)
Rearrange to isolate \(y\): \(y=\sqrt[3]{x}+6\)
Rewrite to match form \(a\cdot\sqrt[3]{bx+c}+d\): \(y=1\cdot\sqrt[3]{1x+0}+6\)

Answer:

For the first question: yes
For the second question: \(1\cdot\sqrt[3]{1x + 0} + 6\) (or simplified as \(\sqrt[3]{x}+6\))