Pre calculus inverse relations and functions

Question

what is the inverse for f(x) = 4/(x-4)^2

Answer 1

The trick is to, instead of looking at the function as f(x)=4/(x-4)^2, look at it as f(x)=y=4/(x-4)^2. From here you should try to solve for x in terms of y to find the inverse function. To do this we first multiply both sides by (x-4)^2 to give us y(x-4)^2=4. Now we divide both sides by y to reach (x-4)^2=4/y. Now we take the square root of both sides to show that sqrt((x-4)^2)=x-4=sqrt(4/y)=2/sqrt(y). Now we simply add 4 to both sides of the equation to see that x=2/sqrt(y)+4=f^-1(y). Since the variable y is arbitrary this means that f^-1(x)=2/sqrt(x)+4! :) You can then check this by showing that f^-1(f(x))=f(f^-1(x))=x, the definition of an inverse function.

Answer 2

what is the inverse for f(x) = 4/(x-4)^2

You have y = f(x) = 4/(x-4)^2

Th e idea is to solve the equation for x

y = 4/(x-4)^2

y* (x-4)^2 = 4

(x-4)^2 = 4/y

take the square root of both sides:

x - 4 = (4/y )^1/2 = 2*y^ -1/2 = 2/ sqrt(y)

So x = (2/ sqrt(y) )+ 4 (swap x and y to get the inverse function in standard format.....)

To check, use a graphing calculator to plot both this and the original equation. Here is a good graphing utility:

https://www.desmos.com/calculator

If you plot the functions, and then plot the line Y = X, you can see how an inverse funciton is the result of mirroring the th function about that line.