Answer to Question #298182 in Calculus for shahana

Question

find the inverse of  f(x) = 5/2 − x, x < 2 ;

1/x, x ≥ 2

Answer 1

Substitute $y$ for $f(x)$

$$y= \begin{cases} 5/2-x, x<2 \\ 1/x, x\ge2 \end{cases}$$

Interchange the variables $x$ and $y$

$$x= \begin{cases} 5/2-y,& y<2 \\ 1/y, & y\ge2 \end{cases}$$

Solve for $y$

$$y= \begin{cases} 1/x,& 0<x\le1/2 \\ 5/2-x, & x>1/2 \end{cases}$$

Substitute $f^{-1}(x)$ for $y$

$$f^{-1}(x)= \begin{cases} 1/x,& 0<x\le1/2 \\ 5/2-x, & x>1/2 \end{cases}$$