An algebraic function is a type of equation that uses mathematical operations.

An equation is a function if there is a one-to-one relationship between its x-values and y-values.

## Algebraic Functions

An **algebraic function** is a function that involves only algebraic operations, like, addition, subtraction, multiplication, and division, as well as fractional or rational exponents. Think of an algebraic function as a machine, where real numbers go in, mathematical operations occur, and other numbers come out.

We call the numbers going into an algebraic function the input, *x*, or the **domain**. Any number can go into a function as long as it is not divided by zero or does not produce a negative square root. A function can preform many mathematical operations with a **domain** as long as the range is one value for each domain used. We call the numbers coming out of a function the output, *y,* or the **range**. Remember, one value in, one value out.

There are many different types of algebraic functions: linear, quadratic, cubic, polynomial, rational, and radical equations. In this next part of the lesson, we’ll learn about a couple of different methods we can use to identify them.

## Tables

One way of identifying an algebraic function is through the use of a table, which can show us if there is one domain and one range. Sometimes functions add to the domain to get the range, like *x* + 2.

Sometimes functions multiply the domain to get the range, like 3*x*. Functions may also subtract or divide the domain or use a combination of operations to produce the range. As long as the rule of ‘one in/one out’ is kept in place, the function exists.If an algebraic function says to add two to the domain, we can create a table to show the function:

Here is an example of a graph that is not a function.

## Examples of Functions

As we said at the beginning of the lesson, there are many types of functions, such as the quadratic function and the cubic function. Let’s start with a quadratic function.The quadratic function: *g(x)* = *x*^2 – 3.

First, let’s create a table.The domain can be any real number, this is why the *x*-value or domain is called the **independent variable**. Here we’ll use -2, -1, 0, 1, and 2 for the domain.

The range or *y*-value is called the **dependent variable** because it depends on what we use for the *x*-term.

Writing the numbers in our table as ordered pairs, we have: (-2,1), (-1, -2), (0, -3), (1, -2), (2, 1). Did you notice how we repeated some of the range values? Is this acceptable? YES, because each ordered pair has one *x*-value and one *y*-value. The *y*-values can be repeated as long as the *x*-values are not repeated.

Remember one in/one out!Now let’s graph this function and use the vertical line test to make sure we still have an algebraic function.

We call this function a quadratic function because it has a squared term, *x*^2.Now let’s take a look at a cubic function: *h(x)* = *x*^3 + *x* – 2 and use the following values for the domain: -2, -1, 0, 1, and 2.

Writing these values as ordered pairs we have: (-2, -12), (-1, -4), (0, -2), (1, 0), (2, 8).On a graph our ordered pairs will look like this.

After applying the vertical line test, we can see this is truly a function!

## Lesson Summary

An **algebraic function** is an equation that allows one to input a **domain**, or *x*-value and perform mathematical calculations to get an output, which is the **range**, or *y*-value, that is specific for that particular *x*-value.There is a one in/one out relationship between the domain and range.There are several ways to show a function:

- Use a table to calculate the range values from the randomly chosen domain values.
The domain is the independent variable. The range is the dependent variable.

- List the domain and range as ordered pairs, (
*x*,*y*). - Show the function as a graph. If a vertical line can pass through any part of the graph and only touch at one point, then the graph is a function. If the vertical line crosses two points, then the graph is not a function.