Relations and Functions
A relation is a rule that takes an input from a set (called the domain) and gives one or more outputs of another set (called the range).
Examples
A function is a relation such that for every input there is exactly one output.
Examples: Examples 2 and 3 above are functions, whereas examples 1 and 4 are not functions.
Determining the Domain of a Function Rule 1: The domain of a polynomial is the set of all real numbers.
Example: For f(x) = 3x^{2} + 2x - 1 the domain is the set of all real numbers.
Rule 2: The domain of a rational function (poly)/(poly) is the set of all real numbers except where the denominator is 0.
Example: For
x - 1 the domain is all real numbers except where x = -1.
Rule 3: The domain of a square root function is all real numbers that make the inside of the square root positive. Example: Find the domain of
Solution: We set up the inequality x^{2} - x - 6 > 0 and use our steps of quadratic inequalities to solve. Factoring we get (x - 3) (x + 2) > 0 Putting -2 and 3 on a number line, gives three regions. The table shows
Hence the solution is (-,-2] U [3,)
The Vertical Line Test If every vertical line passes through the graph at most once then the graph is the graph of a function. Example The Circle is not the graph of a function since there are vertical lines that cross it twice.
Back to the Functions Home Page Back to the Intermediate Algebra (Math 154) Home Page |