Inverse Functions

Question

If f(x)=(x+2)/(x+9), then f^-1(-4)=?

Answer 1

Use y instead of f^-1(x), it's less messy

 

x = (y + 2)/(y + 9)

x(y + 9) = y + 2

xy + 9x = y + 2

xy - y = 2 - 9x

y(x - 1) = 2 - 9x

y = (2 - 9x) / (x - 1)

 

Now substitute -4 for x

 

Answer 2

If f(x)=(x+2)/(x+9), then f^-1(-4)=?

 

Here are the steps in finding f^-1(x)

 

1) Replace f(x) with y

    y = (x+2)/(x+9)

 

2) swap x and y in the equation

    x = (y+2)/(y+9)

 

3) solve for y

    y + 2 = x(y+9)

    y + 2 = xy + 9x

    y - xy = 9x - 2

    y(1 - x) = 9x - 2

    y = (9x-2)/(1-x)

 

4) replace y with f^-1(x)

   f^-1(x) = (9x-2)/(1-x)

 

f^-1(-4) = (9(-4)-2)/(1-(-4))

         = -38/5 = -7 3/5 = -7.6