College Algebra, Chapter 3, 3.7, Section 3.7, Problem 66

$\displaystyle f(x) = 2 - \frac{1}{2} x$ is a one-to-one function. (a) Find the inverse of the function. (b) Graph both the function and its inverse on the same screen to verify that the graphs are reflections of each other in the line $y = x$.

a.) To find the inverse, we set $y = f(x)$.


$
\begin{equation}
\begin{aligned}

y =& 2 - \frac{1}{2} x
&& \text{Solve for $x$, add } \frac{1}{2} x \text{ and subtract } y
\\
\\
\frac{1}{2} x =& 2 - y
&& \text{Multiply both sides by } 2
\\
\\
x =& 2 (2 - y)
&& \text{Interchange $x$ and $y$}
\\
\\
y =& 2(2 - x)
&&

\end{aligned}
\end{equation}
$


Thus, the inverse of $\displaystyle f(x) = 2 - \frac{1}{2} x$ is $f^{-1} (x) = 2 (2 - x)$.

b.)