Answer
one-to-one function
inverse: $f^{-1}(x)=-2x-4$
Work Step by Step
Some of the ordered pairs of the given function, $
f(x)=-\dfrac{1}{2}x-2
$, are $
\left\{(-4,0)(-2,-1),(0,-2),(2,-3),(4,-4),...\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*}
y&=-\dfrac{1}{2}x-2
\\&\Rightarrow
x=-\dfrac{1}{2}y-2
&(\text{interchange $x$ and $y$})
\\\\&
x+2=-\dfrac{1}{2}y
&(\text{solve for $y$})
\\\\&
(-2)(x+2)=\left(-\dfrac{1}{2}y\right)(-2)
\\\\&
-2x-4=y
\\&
y=-2x-4
.\end{align*}
Hence, the inverse is $
f^{-1}(x)=-2x-4
$.
