Question 126726
Find the inverse of:
{{{y = sqrt(x)+9}}} This could be written as:{{{f(x) = sqrt(x)+9}}}
To find the inverse of a function, interchange the x and y and solve for y.
{{{x = sqrt(y)+9}}} Now solve for y by subtracting 9 from both sides.
{{{x-9 = sqrt(y)}}} Square both sides.
{{{(x-9)^2 = y}}} Simplify.
{{{y = x^2-18x+81}}} Now replace the y with {{{f^(-1)(x)}}}:
{{{f^(-1)(x) = x^2-18x+81}}} This is the inverse function of {{{f(x) = sqrt(x)+9}}}. And...yes, the inverse is a function!
To verify this, use the following rule: {{{f(f^(-1)(x)) = x}}}
In this case:
{{{f(f^(-1)(x)) = f(x^2-18x+81)}}} Replace every x with{{{sqrt(x)+9}}} and simplify:
{{{highlight((sqrt(x)+9)^2)-highlight(18(sqrt(x)+9))+81}}} Simplify.
{{{highlight(x+18sqrt(x)+81)-highlight(18sqrt(x)-162)+81}}} Combine like-terms.
{{{x+highlight(18sqrt(x)-18sqrt(x))+highlight(81-162+81)}}}={{{x}}}