How do you find the inverse of y=x^2+12xy=x2+12x?

1 Answer
Johnson Z.
Dec 9, 2016

f^-1(x)=+-sqrt(x+36)-6f−1(x)=±√x+36−6

Explanation:

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

11. Replace f(x)f(x) with yy if it hasn't been done so already.
22. Swap xx and yy.
33. Solve for yy.
44. Replace yy with f^-1(x)f−1(x).

Using these four steps, let us find the inverse of y=x^2+12xy=x2+12x.

Starting with,

y=x^2+12xy=x2+12x

Notice how xx 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 xx only appears once in the equation.

Completing the square,

y=x^2+12x+(12/2)^2-(12/2)^2y=x2+12x+(122)2−(122)2

y=(x+6)^2-(12/2)^2y=(x+6)2−(122)2

y=(x+6)^2-36y=(x+6)2−36

Since the function is already denoted by the variable yy, we go onto swapping xx and yy.

x=(y+6)^2-36x=(y+6)2−36

Solving for yy,

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

+-sqrt(x+36)=y+6±√x+36=y+6

y=+-sqrt(x+36)-6y=±√x+36−6

Replacing yy with f^-1(x)f−1(x),

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

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