Function Basics MathBitsNotebook.com Topical Outline | Algebra 2 Outline | MathBits' Teacher Resources Terms of Use   Contact Person: Donna Roberts

You have worked with relations and functions in the past. Let's refresh our memories and add a few more details.

A relation is simply a set of input and output values, represented in ordered pairs. It is a relationship between sets of information.

Any set of ordered pairs may be used in a relation.
No special rules need apply to a relation.

Consider this example of a relation: The relationship between eye color and student names. (x,y) = (eye color, student's name) Set A = {(green,Steve), (blue,Elaine), (brown,Kyle), (green,Marsha), (blue,Miranda), (brown, Dylan)} Notice that the x-values (eye colors) get repeated.

The scatter plot and the graph, shown below, are also examples of relations.
The thing to notice about them is that they also allow for one x-value
to have more than one corresponding y-value.
Points such as (1,1) and (1,2) can BOTH belong to the same relation.

Relation: {(1,1),(1,2),(3,3),(4,4),(5,5),(5,6),(6,4)}

Relation: ; allows for points such as (2,1.414) and (2,-1.414).

If you add a "specific rule" to a relation, you get a function.

A function is a set of ordered pairs in which each x-element has only ONE y-element associated with it.

While a function may NOT have two y-values assigned to the same x-value, it may have two x-values assigned to the same y-value. NOT OK for a function: {(5,1),(5,4)} OK for a function: {(5,2),(4,2)} Function: each x-value has only ONE y-value!

If we remove duplicate eye colors, the eye color example will be a function: (x,y) = (eye color, student's name) Set B = {(blue,Steve), (green,Elaine), (brown,Kyle)} Set B is a function.

If we remove (1,2) and (5,6), we will have a function. Function: {(1,1),(3,3),(4,4),(5,5),(6,4)}

If we change the ± sign to just a + sign, we will have a function. Function:

Notice that vertical lines on the graphs make it clear if an x-value had more than one y-value.
If the vertical lines intersected the graph in more than one location, we had a relation, NOT a function.

Vertical line test for functions: Any vertical line intersects the graph of a function in only ONE point.

Given that relation A = {(4,3), (k,5), (7,-3), (3,2)}. Which of the following values for k will make relation A a function?     a) 3      b) 4      c) 6

Solution: Choice c. The x-values of 3 and 4 are already used in relation A. If they are used again (with a different y-value), relation A will not be a function.

Which of the following graphs represents a function?

a)

b)

c)

d)

Solution: Choice b. A vertical line drawn on this graph will intersect the graph in only one location, making it a function. Vertical lines on the other three graphs will intersect the graph in more than one location, or as in part a, will intersect in an infinite number of points (all points).

Calculators graph functions!

By solving for "y =", you are actually identifying a "function".

If you can solve an equation for "y =", then the equation is a "function".
4x + 2y = 10
2y = -4x + 10
y = -2x + 5 (a function!)
This equation can now be entered into a graphing calculator for plotting.

Most calculators (including the TI-84+ series) can only handle graphing functions.
The equation (function) must be in "y = " form before you can enter it in the calculator.

BUT ... what about y² = x ?
If we solve for "y =", we get , which we saw, at top of this page, was not a function.
We cannot graph this on our calculator as a single entity, since there is no key for "±".
We were not able to solve this equation for a unique (only one) "y =" equation.
We actually have two "y =" equations: and .
(Yes, the graphing calculator can graph these equations separately to form the graph.
But the combined graphs will be a relation, not a function.)
The lack of a unique "y =" equation means that y² = x is not a function.