Tutorial

Don't worry. Although this is in the title, it won't be a difficult section. A relation is basically any group of ordered pairs. For example, look at the points below.

In Algebra II, the list of 'x' numbers is called the domain (sometimes called the independent variables or inputs). Likewise, the list of 'y' numbers is called the range (sometimes called dependent variables or outputs).

After you group the numbers together, you can connect them like this:

Functions are similar to relations, except that no two ordered pairs have the same first number. That means that a number in the domain cannot point to two numbers in the range(therefore, the diagram above is NOT a function). Another way to find if it is a function, works on a graph. What you do is you take a vertical line and see if it crosses more than one point on the line at once. The one below is not a function.

Functional notation is represented by the equation f(x)=ax+b. f(x) does not mean 'f' times 'x'. It means the function of 'x'. Sometimes this is interchangeable with y. Let's say you have f(x)=2x+3, but the dreaded math textbook asks you to solve f(4)=2x+3. All you do is substitute x for 4 to get an answer of 11. Want to know what else you get out of that? The ordered pair (4,11).

'f' may be substituted for 'g' or 'h'. They may all be used if you have to graph multiple functions. I bet you already know a form of this. The area of a square is represented by this equation:

A=area, r=radius

RULE: Every 'y' value must have only one 'x' value.

You may remember that we use the vertical line test to see if a line was a function. To see if a line is a 1 to 1 function, we use the horizontal line test. Although the line below is a function, it is not 1 to 1.

RULE: Every 'y' value has an 'x' value.

That means that if you use the horizontal line test, your line should hit at least one point on the graph at any point.

One way to remember them is:

1-1: the horizontal line hits the graph at most once at any point

Onto: the horizontal line hits the graph at least once at any point

Let's first plot some different functions.

- f(x)=x
^{2} - g(x)=2
^{x} - h(x)=x
^{3}-3x^{2}-5x+6 - j(x)=x
^{3}

g(x)=Appears to be neither "1 to 1" nor "onto"

h(x)Appears to be "onto", but not "1 to 1"

j(x)=Appears to be both "1 to 1" and "onto"

We can also make a chart. If there is "Yes" in 0 and/or 1, it's 1-1. If it's 1 or more, it's onto.

Function | Hits 0 spots | Hits 1 spot | Hits 2 spots | Hits 3+ spots | 1-1 | Onto |

f(x) | Yes | Yes | Yes | - | - | - |

g(x) | Yes | Yes | - | Yes | - | - |

h(x) | - | Yes | Yes | - | - | Yes |

j(x) | - | Yes | - | - | Yes | Yes |

As you can see, a function can be neither or both too.

P.S. This bad drawing is why I copy and paste pictures usually.

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- Relations
Domain and Range

Functions

Functional Notation

Linear Functions (COMING SOON)

Script Writer: Bryce Schoenherr

Head Mathematician: {PRIVATE}

Mathematicians: Bryce Schoenherr