Finding an inverse function

Question

Determine whether function has an inverse function, if it does, find the inverse function.

f(x) = x⁴

f(x) = 3x + 5

Answer 1

To determine whether or not a function has an inverse, we can simply look at the graph and see whether it passes the vertical line test. This is something you've likely done before, but if not, this is accomplished by seeing that for every value of x, there is only one value of y.

Now to actually solve for the inverse, you can start by replacing f(x) with y. Then, solve for x. For the first problem, f(x) = x⁴, we start with y = x⁴. Next, because there are no other terms, we can take the fourth root of y which can be written as y^1/4. Because x is completely isolated, we now have the inverse function!

x = y^1/4 = f^-1(x)

The second problem is even easier. Starting off, we have y = 3x + 5. Solving for x only requires us to subtract both sides of the equation by 5, and then divide by 3. This gives us the following answer.

x = (y - 5) / 3 = f^-1(x)

Answer 2

All functions have inverses. Some, not all, functions have inverses that are functions.

f(x) = x⁴ has the inverse y = ⁴√x, yet this inverse is not a function

f(x) = 3x + 5

x = 3f^-1(x) + 5

x - 5 = 3f^-1(x)

(x - 5) / 3 = f^-1(x). This inverse is a function.