Now that we know about the domain and range, we can update our list of famous functions.

In Section 1-2, you saw a variety of famous functions. Now that we have learned more about properties of functions, we can update our knowledge of those famous functions. We will go through the list of famous functions from before and point out where each function might have properties we’ve discussed.

Linear Functions

Recall that the graph of a linear function is a line.

In general, linear functions can be written as where and can be any numbers. We learned that represents the slope, and is the -coordinate of the -intercept. You can play with changing the values of and on the graph using Desmos and see how that changes the line.

Note that a linear function defined by with is odd. If , then is periodic, since it is constant. Furthermore, constant functions are always even.

Additionally, if , then a linear function is one-to-one, and therefore invertible. We summarize this information in the table below.

Note that any real number can be plugged into , so the domain of linear functions is . Unless , we can find a such that , so the range of linear functions with is . If , then the only output of the linear function is , so its range is .

Quadratic Functions

Recall that the graph of a quadratic function is a parabola.

In general, quadratic functions can be written as where , , and can be any numbers. You can play with changing the values of , , and on the graph using Desmos and see how that changes the parabola.

Note that for a quadratic function defined by , if , then is even. In general, quadratic functions are not one-to-one, odd, or periodic, except in cases where , in which we’re actually dealing with a linear function.

Note that any real number can be plugged into , so the domain of quadratic functions is . In Chapter 4, we saw that all quadratic functions have a vertex form , where the vertex is at . If , all points above the vertex, that is are in the range of the quadratic, and if , all points below the vertex, that is are in the range of the quadratic.

We summarize this information in the table below.

Absolute Value Function

Another important type of function is the absolute value function. This is the function that takes all -values and makes them positive. The absolute value function is written as

Notice that the absolute value function is even. Is it one-to-one? The fact that it’s even tells us that it is not, since for all . We summarize this information in the table below.

Note that any real number has an absolute value, so the domain of the absolute value function is . Furthermore, by looking at the graph, we can see that all non-negative numbers are in the range of the absolute value function.

Square Root Function

Another famous function is the square root function,

The square root function is one-to-one. Negative inputs are not valid for the square root function, so it is neither even, odd, nor periodic. We summarize this information in the table below.

Note that only non-negative numbers have square roots, so the domain of the square root function is . Furthermore, by looking at the graph, we can see that all non-negative numbers are in the range of the square root function. Algebraically, we can say that for any non-negative , , so is in the range of the square root function.

Exponential Functions

Another famous function is the exponential growth function,

Here is the mathematical constant known as Euler’s number. .

In general, we can talk about exponential functions of the form where is a positive number not equal to . You can play with changing the values of on the graph using Desmos and see how that changes the graph. Pay particular attention to the difference between and .

Notice that exponential functions are one-to-one, and therefore invertible. However, they are neither even, odd, nor periodic.

Note that the domain of the exponential functions is . Furthermore, by looking at the graph, we can see that all non-negative numbers are in the range of the exponential functions.

We summarize this information in the table below.

Logarithm Functions

Another group of famous functions are logarithms.

In general, we can talk about logarithmic functions of the form where is a positive number not equal to . You can play with changing the values of on the graph using Desmos and see how that changes the graph. Pay particular attention to the difference between and .

Notice that logarithms are neither even, odd, nor periodic. However, they are one-to-one, and therefore invertible. It turns out that the inverse of a logarithm is an exponential function, and vice versa!

Note that since the logarithm is the inverse of the exponential, the domain of the logarithms is the range of the exponentials: . Furthermore, the range of the logarithms is the range of the exponentials: .

We summarize this information in the table below.

Sine

Another important function is the sine function,

This function comes from trigonometry. In the table below we will use another mathematical constant, (“pi” pronounced pie). .

As mentioned earlier, the sine function is odd and periodic with period . Since it is periodic, however, it cannot be one-to-one, since its values repeat.

Note that the domain of the sine function is . Furthermore, by looking at the graph, we can see that its range is .

We summarize this information in the table below.

In general, we can consider . You can play with changing the values of and on the graph using Desmos and see how that changes the graph.

Cosine

A function introduced in Section 3-2 is the cosine function,

As with sine, the cosine function comes from trigonometry. In the table below we will again use .

As mentioned earlier, the cosine function is even and periodic with period . Since it is periodic, however, it cannot be one-to-one, since its values repeat.

Note that the domain of the cosine function is . Furthermore, by looking at the graph, we can see that its range is .

We summarize some information in the table below.

In general, we can consider . You can play with changing the values of and on the graph using Desmos and see how that changes the graph.