Answer
$(a)$ $f(x)=(x-2)^3$
$(b)$
$f(-1)=-27$
$f(0)=-8$
$f(1)=-1$
$f(2)=0$
$f(3)=1$
$f(4)=8$
$(c)$ See the image below.
$(d)$ We can visually apply the Horizontal Line Test. As we can see, there is no horizontal line that crosses the graph $2$ times. So, we can assume that the function has an inverse.
$(e)$ $f^{-1}=\sqrt[3]{x}+2$
Work Step by Step
$(a)$ The verbal description can be written as (Each step is shown):
$f(x)=x$
$f(x)=x-2$
$f(x)=(x-2)^3$
$(b)$ We will simply input the values and calculate corresponding output:
$f(-1)=(-1-2)^3=(-3)^3=-27$
$f(0)=(0-2)^3=(-2)^3=-8$
$f(1)=(1-2)^3=(-1)^3=-1$
$f(2)=(2-2)^3=0^3=0$
$f(3)=(3-2)^3=1^3=1$
$f(4)=(4-2)^3=2^3=8$
$(c)$ We can simply plot the points using the results in $(b)$
See the image above.
$(d)$ We can visually apply the Horizontal Line Test. As we can see, there is no horizontal line that crosses the graph $2$ times. So, we can assume that the function has an inverse.
$(e)$ We have to first write the function in terms of $y$ and $x$, then replace $x$ by $y$ and vice versa and at last simplify.
$y=(x-2)^3$
$x=(y-2)^3$
$y-2=\sqrt[3]{x}$
$y=\sqrt[3]{x}+2$
$f^{-1}=\sqrt[3]{x}+2$
