Functions, Equations, and Graphs

Chapter #1

Relations

A set of ordered pairs of input and output values

• Can be represented as ordered pairs, mapping diagrams, tables, or graphs

Function

• A relation in which one element of the domain (x values) corresponds with exactly one element of the range (y values).

  • For every x-value, there is only one y-value.

  • Two different x-values can share the same y-value.

• Mapping diagrams → only one arrow from each element of the domain.

• Can use the vertical line test for graphs.

  • If a vertical line test is drawn through the graph and intersects at one point, it is a function.

Function rule

An equation that represents an output value in terms of an input value.

• Example:

  • y (output) = 3x + 2 (input is x)

Function notation

A way to write a function rule

• f(x) = 3x + 2

  • Example:

  • f(1) = 3(1) + 2

    f(1) = 5

• Independent variable(x): input

• Dependent variable(f(x)): output. It’s value depends on the input.

Direct Variation

• A relationship where the ratio of values is constant.

• Can write a formula for direct variation functions as:

  • K = y/x or y = kx

  • x → input values

  • y → output values

• The formula k = y/x means that the ratio of output/input equals the constant k (constant of variation)

• On a graph, direct variation must go through the origin.