Question

The rational function f( 2/-2×-5)

A .Find f¹(×)

B. Find the equation of the line of symmetry of f¹(×) and f(×).

Answer 1

A.

$$f(x)=\dfrac{2}{-2x-5}$$

$$-2x-5\not=0=>x\not=-2.5$$

$$y=\dfrac{2}{-2x-5}$$

Interchange $x$ and $y$

$$x=\dfrac{2}{-2y-5}$$

Solve for $y$

$$2y+5=-\dfrac{2}{x}$$

$$y=-\dfrac{1}{x}-\dfrac{5}{2}$$

$$y=-\dfrac{2+5x}{2x}$$

Replace $y$ with $f^{-1}(x)$

$$f^{-1}(x)=\dfrac{-2-5x}{2x}$$

B.

The line of symmetry of $f^{-1}(x)$ and $f(x)$ is $y=x.$