Question

How do you find the inverse of `f(x)=(x-3)/(x+2)` and graph both f and `f^-1`?

Answer 1

Switch the x and y values to find the inverse.

`y = (x - 3)/(x + 2)`

The inverse is given by

`x= (y - 3)/(y + 2)`

Solve for `y` now:

`x(y + 2) = y - 3`

`xy + 2x = y - 3`

`2x + 3 = y - xy`

`2x + 3 = y(1 - x)`

`(2x + 3)/(1 - x) = y`

The inverse, `f^-1(x)`, is given by `f^-1(x) = (2x + 3)/(1 - x)`.

The function can be graphed using knowledge of asymptotes, invariant points and intercepts. Prepare a table of values for `f(x)`. Recall that `f^-1(x)` is simply a transformation of `f(x)` over the line `y = x`, so `f^-1(x)` has a table of values where `x` and `y` are inverted relative to `f(x)`.

For example, if the point `(2, 3)` belongs on the graph of `f(x)`, the point `(3, 2)` belongs on `f^-1(x)`.

Here is the graph of `f(x)`:

Here is the graph of `f^-1(x)`:

Hopefully this helps!