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

1 Answer
Dec 9, 2016

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) | )))