The given function is,
[tex] f(x)= \frac{2x-1}{x+5} [/tex].
Or, [tex] y= \frac{2x-1}{x+5} [/tex]
To find the inverse of a function, first step is to switch x and y. Therefore,
[tex] x= \frac{2y-1}{y+5} [/tex]
Next stpe is to solve the above equation for y t get the inverse f f(x). Hence, multiply each sides by y + 5 to get rid of fraction form. So,
[tex] x*(y+5)= \frac{2y-1}{y+5}*(y+5) [/tex]
xy + 5x = 2y - 1
xy = 2y - 1 - 5x Subtract 5x from each sides to isolate y.
xy - 2y = -5x - 1 Subtract 2y from each sides.
y(x - 2) = -5x -1 Take out the common factor y.
[tex] \frac{y(x-2)}{(x-2)} =\frac{(-5x-1)}{(x-2)} [/tex] Divide each sides by x-2.
[tex] y=\frac{(-5x-1)}{(x-2)} [/tex]
So, [tex] f^-1(x)=\frac{-5x-1}{x-2} [/tex]
Now when we will compare [tex] f^-1(x)=\frac{-5x-1}{x-2} [/tex] with [tex] f^-1(x)=\frac{ax+b}{cx+d} [/tex] then we will get a = -5, b = -1, c = 1 and d=-2
So, [tex] \frac{a}{c} =\frac{-5}{1} =-5 [/tex]
Hope this helps you!
