Answer to Question #298182 in Calculus for shahana

Question #298182

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

1/x, x ≥ 2


1
Expert's answer
2022-02-16T15:24:19-0500

Substitute yy for f(x)f(x)


y={5/2x,x<21/x,x2y= \begin{cases} 5/2-x, x<2 \\ 1/x, x\ge2 \end{cases}

Interchange the variables xx and yy


x={5/2y,y<21/y,y2x= \begin{cases} 5/2-y,& y<2 \\ 1/y, & y\ge2 \end{cases}

Solve for yy


y={1/x,0<x1/25/2x,x>1/2y= \begin{cases} 1/x,& 0<x\le1/2 \\ 5/2-x, & x>1/2 \end{cases}

Substitute f1(x)f^{-1}(x) for yy


f1(x)={1/x,0<x1/25/2x,x>1/2f^{-1}(x)= \begin{cases} 1/x,& 0<x\le1/2 \\ 5/2-x, & x>1/2 \end{cases}

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