Algebraic Function: Definition & Examples

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  • 0:02 Algebraic Functions
  • 1:14 Tables
  • 2:23 Graphs
  • 3:06 Vertical Line Test
  • 3:30 Examples of Functions
  • 5:26 Lesson Summary
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Lesson Transcript
Instructor: Ellen Manchester
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.

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 Table

As you can see, for every domain, we have one range. These pairs of x values- and y-values are called ordered pairs because we put them in order (x,y).

We can also turn our table into ordered pairs to show a function: (1,3), (4,6), (-2,0) and (-3,-1) where there is one x-value for every one y-value.


We can also use graphs to identify functions by plotting ordered pairs onto a Cartesian Coordinate System, where the x-values are on the horizontal line and the y-values are on the vertical line. Where the ordered pairs meet is where the point is graphed. If we plot the points, we end up with a straight line, so the function, x + 2, is considered a linear function and can be written in functional notation as f(x) = x + 2. The f(x) is just another way to write y, which we call the f-function. It is a way for us to identify the different functions, instead of calling them all y = ...

Linear Function
Linear Function

Vertical Line Test

We know a graph is a function if it can pass the vertical line test. In this test, if we place a vertical line anywhere on a graph, it will cross in only one place. If a vertical line crosses in two places on a graph, it is in conflict with the one in, one out rule. So, it is not a 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

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