An with a straight line, so the

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.

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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.

Function Machine
Function Machine


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 3x. 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:

Linear Function
Vertical Line Test on Linear Function
Vertical Line Test on Linear Function

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

Not a Function
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.

Quadratic function table

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.

Quadratic function graph

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.

Cubic Function Table

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.

Cubic Function Graph

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

Cubic Function - Vertical Line Test

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:

  1. 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.

  2. List the domain and range as ordered pairs, (x,y).

  3. 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.

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