When given functions f(x) and g(x), new functions can be formed new using composition and mathematical operations such as addition, subtraction, multiplication and division. We can also create new functions by finding inverses. Suppose we have a function f(x) where exactly one member of the domain corresponds to one member of the range. Reverse the function so that each member of the range of f(x) now becomes the domain and the domain now becomes the range. Such a correspondence is called the inverse of f(x).

Suppose the domain of a function is (2,4,6,8,10) and the range is (-2, -2, -3, -4, -5). In this case, reversing will not produce a function because there are two numbers in the range for a single number in the domain. Notice after the reversing process -2 in the domain corresponds to both 2 and 4 in the range. Therefore, for a function to have an inverse that is also a function, each member of the domain must determine exactly one output of the range. Such a function is called a one-to-one function.

We can determine if a graph is a function if each vertical line intersects the graph only once. Similarly, we can determine if a function is one-to-one if each horizontal line intersects the graph only once. This is known as the horizontal line test.

For a function to be one – to – one, we can either graph the function and see if it passes the horizontal line test or input values for x to see if there are more than one with the same output. We can tell that this function is one – to – one without graphing or inputting numbers because the function is a line and since it’s not a horizontal line, it passes the horizontal line test. If you cannot see that, we can graph it using the slope and y- intercept. Recall that the equation of a line is y = mx + b, where m is the slope and b is the y- intercept. So this is a line with a slope of 3 and y- intercept of 4. Notice the graph of the line below passes the horizontal line test.

f(x) = x2

This function is not one – to – one and we can tell this by inputting values for x. Since x will be squared, all outputs for x will be positive.

Let x = 1, then f(x) = 1.

Let x = -1, then f(x) = 1.

Notice the same output of 1 for two different inputs. This shows that the function is not one – to – one. Graphing the function will also show it is not one – to – one since it doesn’t pass the horizontal line test.

Now that we understand the concept of inverse functions, we need to know how to find the equation of the inverse of a function. Notice that the domain of the function becomes the range of the inverse function and the range of the function becomes the domain of the inverse function. That means we must interchange the coordinates to determine the inverse function. The inverse of f(x) is denoted by the symbol f -1(x).

**Examples:** Determine whether or not the following functions are one – to – one. If they are one – to – one, find its inverse.

1. f(x) = -4x + 2

Since the function is linear, we know it’s one – to – one. Therefore to find the inverse we first substitute f(x) with y. That gives us y = -4x + 2.

Next we interchange x and y to get x = -4y + 2. Now solve the equation for y.

x – 2 = -4y + 2 – 2

x – 2 = -4y

y = -(x – 2)/4

Finally, replace y with f -1(x) to get

f -1(x) = -(x – 2)/4

2. f(x) = (1/3)x3

We can graph this function by substituting a few values for x to determine the direction of the curve and then draw a smooth curve through the points. Or we can use a graphing calculator to determine the graph. We can see that that f(x) increases as x increases.

Next, we find the inverse.

f(x) = (1/3)x3

y = (1/3)x3

x = (1/3)y3

3√(3x) = 3√(y3)

3√(3x) = y

f -1(x) = 3√(3x)

This guide should help assist anyone having trouble understanding inverses and finding inverses of functions.