Five Representations

Relation: any set of ordered pairs

Examples A = {(2,5), (-7,4.8), (4,12), (2,7)} B = {(9,0), (4,5), (8.1,0.23), (4,5)} C = {(1, red), (2, red)}

Function: a set of ordered pairs that maps each x-value to exactly one y-value

In the examples above, B is a function while A is not. Can you explain why this is true?

Typically, we use five representations to describe functions: 1) a set/listing of ordered pairs, 2) a graph, 3) a table of values, 4) an arrow diagram, and 5) a rule/formula.

F = {(3,4.5), (-9,7), (0.8,-12)} domain D = {3, -9, 0.8} and range R = {4.5, -12, 7}

The graph would be the set of three points mapped onto the Cartesian plane in the usual way. A graph is a picture of the function. Recall that the vertical line test (VLT) can be used to test a graph to determine if it is a function. The criteria that must be met is that the vertical line pass through only one point on the graph at a time.

The table of values would be as follows:

x	y
3	4.5
-9	7
0.8	-12

The table tells us that F(-9) = 7 or the value of the function F at -9 is 7.

The arrow diagram is another representation of the function that shows how the domain (x) maps into the range (y).

x	y
3 à	4.5
-9 à	7
0.8 à	-12

Notice that in an arrow diagram for a function, each x-value has exactly one arrow coming out of it to the same y-value.

The last representation is the rule or formula. Suppose function g(x) = 2x + 5. The rule/formula for g is 2xz + 5 because this determines what happens to x to give the related y.

Kinds of Functions

1. linear functions: f(x) = mx + b (highest power on x is 1)

2. quadratic functions: f(x) = ax² + bx + c (highest power on x is 2)

3. exponential functions: f(x) = a(b)^x (x is the power in the function)

4. logarithmic functions: f(x) = a(ln x) + b (x is part of a log/ln function)