For this case we must find the inverse of the following function:
[tex]f (x) = (x-6) ^ 2 + 7[/tex]
For this we follow the steps below:
Replace f (x) with y:
[tex]y = (x-6) ^ 2 + 7[/tex]
We exchange the variables:
[tex]x = (y-6) ^ 2 + 7[/tex]
We solve the equation for "y", that is, we clear "y":
[tex](y-6) ^ 2 + 7 = x[/tex]
We subtract 7 on both sides of the equation:
[tex](y-6) ^ 2 = x-7[/tex]
We apply square root on both sides of the equation to eliminate the exponent:
[tex]y-6 = \sqrt {x-7}[/tex]
We add 6 to both sides of the equation:
[tex]y = \pm \sqrt {x-7} +6[/tex]
We change y by [tex]f ^ {- 1} (x):[/tex]
[tex]f ^ {- 1} (x) = \pm \sqrt {x-7} +6[/tex]
Answer;
[tex]f ^ {- 1} (x) = \pm \sqrt {x-7} +6[/tex]
If it is a inverse function.
