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.

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. 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.

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.

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. 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! 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. 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. 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.