Answer
one-to-one function
inverse: $f^{-1}(x)=\sqrt[3]{x-5}$
Work Step by Step
Some of the ordered pairs of the given function, $
f(x)=x^3+5
$, are $
\left\{(-2,-3)(-1,4),(0,5),(1,6),(2,13),...\right\}
$. Note that every $y$-coordinate from this function is unique. Hence, the given function is a one-to-one function.
To find the inverse, let $y=f(x)$. Then, interchange the $x$ and $y$ variables and solve for $y$. That is,
\begin{align*}\require{cancel}
y&=x^3+5
\\&\Rightarrow
x=y^3+5
&(\text{interchange $x$ and $y$})
\\&
x-5=y^3
&(\text{solve for $y$})
\\&
\sqrt[3]{x-5}=y
\\&
y=\sqrt[3]{x-5}
.\end{align*}
Hence, the inverse is $
f^{-1}(x)=\sqrt[3]{x-5}
$.
