Relations, Equations and Functions

Relations

A relation is a set of ordered pairs, and can be represented by a table, a graph or a mapping.

A table

A graph

Mapping

The solutions of an equation in two variables is an ordered pair that results in a true statement when substituted into the equation.

Linear Equations

A linear equation is the equation of a straight line, and can be written in the standard form as: Ax + By =  C

Another form of a linear equation is:

y = mx + b, where m is the slope of the line on the graph, and b is the point on the y-axis ( y-intercept) where the line crosses the y-axis on the graph of the equation.

Functions

A function is a relation in which each element of the domain ( x-axis values) is paired with exactly one element of the range (y-axis values).

equation  y = 4x + 7

function f(x)  = 4x + 7

Function Only one value of y for all values of x

Not a function More than one value of y for some values of x

To find the value of a function substitute the domain value for x.

Example:

for the function f(x) = 2x - 4, find the value of x + 3.

Substitute (x + 3) for x:

f(x + 3) = 2(x + 3) - 4 = 2x + 2

Difficult Example:

If f(x) = 6x - 3, find the value of 3[ f(x²) - 4 ]

Solve this in 3 steps:

1. find the value of f(x²).

2. Subtract 4 from this value.

3. Multiply the result from step 2 by 3.

Step 1: f(x²) = 6x² - 4

Step 2: 6x² - 4 - 4 = 6x² - 8

Step 3: 3(6x² - 8) = 18x² - 24