Question

given the function $f(x) = \sqrt3{x} + 3$, complete parts a through c.
(a) find an equation for $f^{-1}(x)$.
(b) graph $f$ and $f^{-1}$ in the same rectangular coordinate system.
(c) use interval notation to give the domain and the range of $f$ and $f^{-1}$.
(hint: to solve for a variable involving an nth root, raise both sides of the equation to the nth power, $(\sqrtn{y})^n = y$.)

a) find $f^{-1}(x)$. select the correct choice below and fill in the answer box(es) to complete your choice.
(simplify your answer. use integers or fractions for any numbers in the expression.)
\\( \bigcirc \\) a. $f^{-1}(x) = \square$, $x \geq \square$
\\( \bigcirc \\) b. $f^{-1}(x) = \square$, $x \
eq \square$
\\( \bigcirc \\) c. $f^{-1}(x) = \square$, $x \leq \square$
\\( \bigcirc \\) d. $f^{-1}(x) = \square$, for all $x$

Explanation:

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

Let \( y=\sqrt[3]{x}+3 \).

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

We get \( x = \sqrt[3]{y}+3 \).

Step3: Solve for \( y \)

Subtract 3 from both sides: \( x - 3=\sqrt[3]{y} \).
Raise both sides to the 3rd power: \( (x - 3)^3=y \).

Answer:

D. \( f^{-1}(x)=(x - 3)^3 \), for all \( x \)