Relations and Functions
A function is a relation such that for every input there is exactly one output.
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.
f(x) = 3x2 + 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.
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.
Find the domain of
We set up the inequality
x2 - 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.
The Circle is not the graph of a function since there are vertical lines that cross it twice.