Question

How do you find the inverse of `y=x^2+12x`?

Answer 1

`f^-1(x)=+-sqrt(x+36)-6`

Explanation:

The general steps to finding the inverse of a function are:

`1`. Replace `f(x)` with `y` if it hasn't been done so already.
`2`. Swap `x` and `y`.
`3`. Solve for `y`.
`4`. Replace `y` with `f^-1(x)`.

Using these four steps, let us find the inverse of `y=x^2+12x`.

Starting with,

`y=x^2+12x`

Notice how `x` is found in more than one term. This can create a problem for us when trying to find the inverse. Thus, we can rewrite the equation in vertex form so that `x` only appears once in the equation.

Completing the square,

`y=x^2+12x+(12/2)^2-(12/2)^2`

`y=(x+6)^2-(12/2)^2`

`y=(x+6)^2-36`

Since the function is already denoted by the variable `y`, we go onto swapping `x` and `y`.

`x=(y+6)^2-36`

Solving for `y`,

`x+36=(y+6)^2`

`+-sqrt(x+36)=y+6`

`y=+-sqrt(x+36)-6`

Replacing `y` with `f^-1(x)`,

`color(green)( bar (ul ( | color(white)(a/a) color(black)(f^-1(x)=+-sqrt(x+36)-6) color(white)(a/a) | )))`