Answer
f is one-to-one,
$f^{-1}(x)=-x+2, \quad x\geq 0$
Work Step by Step
In the interval $x \leq 2,$
$(x-2)$ is negative, so
$|x-2|=-(x-2)=-x+2$
$f(x)=-x+2,\quad x \leq 2$
is a linear function, whose graph has negative slope.
Linear functions with slope$\neq$0 are strictly monotonic.
By Th.5.7, they are one-to-one and have inverses.
To find the inverse,
1. swap f(x)=y and x:
$y=-x+2 \qquad , x \leq 2 \Rightarrow \mathrm{y} \geq 0$
$x=-y+2 \qquad , y \leq 2\Rightarrow x \geq 0$
2. Solve for y:
$x=-y+2\qquad /+y-x$
$y=-x+2, \quad x\geq 0$
3. Replace $y $ with $f^{-1}(x):$
$f^{-1}(x)=-x+2, \quad x\geq 0$
