College Algebra, Chapter 3, 3.7, Section 3.7, Problem 68

$\displaystyle f(x) = x^2 + 1, x \geq 0$ 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 =& x^2 + 1
&& \text{Solve for $x$; subtract } 1
\\
\\
x^2 =& y - 1
&& \text{Take the square root}
\\
\\
x =& \pm \sqrt{y - 1}
&& \text{Interchange $x$ and $y$}
\\
\\
y =& \pm \sqrt{x - 1}
&& \text{Apply restrictions, } x \geq 0
\\
\\
y =& \sqrt{x - 1}
&&

\end{aligned}
\end{equation}
$


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

b.)