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Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!

start with:

f(x) = 3^-x + 3.

replace f(x) with y.

you don't have to do this, but it's easier to work with y rather than f(x) and since you set them equal to each other, you can use either one inter-changeably.

your equation becomes y = 3^-x + 3.

replace x with y and y with x to get:

x = 3^-y + 3.

subtract 3 from both sides of the equation to get x - 3 = 3^-y.

this is the same as 3^-y = x - 3.

take the log of both sides of this equation to get log(3^-y) = log(x-3)

since log(3^-y) is the same as -y * log(3), your equation becomes:

-y * log(3) = log(x-3)

divide both sides of the equation by log(3) to get:

-y = log(x-3) / log(3)

multiply both sides of the equation by -1 to get:

y = - log(x-3) / log(3)

the graph of your original equation and your inverse equation look like this:



the inverse equation essentially undoes what the original equation does.

when you're just dealing with x and y, it's not always easy to see this.

in the original equation, y = 3^-x + 3 says that, given the value of A, you can find the value of B by using that equation. The value of A is represented by x and the value of B is represented by y in the equation.

in the inverse equation, however, the equation of y = - log(x-3) / log(3) says that, given the value of B, you can find the value of A. The value of B is represented by x and the value of A is represented by y in the equation.

that is why you get a flip flop in the corresponding coordinate points.

on the graph, you see that the point (-1.736,9.736) in the original equation corresponds with the point (9.736,-1.736) in the inverse equation.

in the original equation, A is represented by x which is equal to -1.736 and B is represented by y which is equal to 9.736.

in the revised equation, B is represented by x which is equal to 9.736 and A is represented by y which is equal to -1.736.

the original equation gives you the value of B given the value of A.
the inverse equation gives you the value of A given the value of B.