Answer
Answer:
$f^{-1}(x) $ = $\frac{5}{2} - x$ when $x \lt 2$
$f^{-1}(x) = \frac{1}{x}$ when $x \geq 2$
Work Step by Step
For the first part, we have: $y = f(x) = \frac{5}{2} - x$. when $x \lt 2$
Thus, $x = \frac{5}{2} - y$.
Now, to get this in the form of an inverse, replace $x$ with $f^{-1}(x) $ on the left-hand side and $y$ with $x$ on the right-hand side.
Thus, $f^{-1}(x) $ = $\frac{5}{2} - x$ when $x \lt 2$.
Now, we have $ y = f(x) = \frac{1}{x}$ when $x \geq 2$.
Thus, $x = \frac{1}{y}$.
Now, to get this in the form of an inverse, replace $x$ with $f^{-1}(x) $ on the left-hand side and $y$ with $x$ on the right-hand side.
Thus, $f^{-1}(x) = \frac{1}{x}$ when $x \geq 2$
We see here that $f^{-1}(x) = f(x) $.
