Option D:
[tex]$y=\frac{5x-2}{x}[/tex]
Solution:
Given function:
[tex]$y=\frac{-2}{x-5}[/tex]
To find the inverse of the function.
Inverse of a function:
If a function f(x) is mapping x to y, then the inverse function of f(x) maps y back to x.
[tex]$y=\frac{-2}{x-5}[/tex]
Interchange the variables x and y.
[tex]$x=\frac{-2}{y-5}[/tex]
Now, solve for y.
Multiply both sides by (y - 5).
[tex]$x(y-5)=-\frac{2}{y-5}(y-5)[/tex]
Cancel the common factors, we get
[tex]x(y-5)=-2[/tex]
Divide by x on both sides.
[tex]$y-5=-\frac{2}{x}[/tex]
Add 5 on both sides.
[tex]$y=-\frac{2}{x}+5[/tex]
[tex]$y=-\frac{2}{x}+\frac{5}{1}[/tex]
To make the denominator same, multiply the 2nd term by [tex]\frac{x}{x}[/tex].
[tex]$y=-\frac{2}{x}+\frac{5x}{x}[/tex]
[tex]$y=\frac{-2+5x}{x}[/tex]
[tex]$y=\frac{5x-2}{x}[/tex]
Option D is the correct answer.
