Foundations Functions, Domain, and Range
Relations & Functions Relation: a set of ordered pairs Domain: the set of x-coordinates Range: the set of y-coordinates When writing the domain and range, do not repeat values.
Relations and Functions Given the relation: {(2, -6), (1, 4), (2, 4), (0,0), (1, -6), (3, 0)} State the domain: D: {0,1, 2, 3} State the range: R: {-6, 0, 4}
Relations and Functions • Relations can be written in several ways: ordered pairs, table, graph, or mapping. • We have already seen relations represented as ordered pairs.
Table 4 2 -1 0 3 3 7 0 -2 -5 Ordered pair {(3, 4), (7, 2), (0, -1), (-2, 2), (-5, 0), (3, 3)} Mapping Graph
Functions • A function is a relation in which the members of the domain (x-values) DO NOT repeat. • So, for every x-value there is only one y-value that corresponds to it. • y-values can be repeated.
Functions • Discrete functions consist of points that are not connected. • Continuous functions can be graphed with a line or smooth curve and contain an infinite number of points.
Do the ordered pairs represent a function? {(3, 4), (7, 2), (0, -1), (-2, 2), (-5, 0), (3, 3)} No, 3 is repeated in the domain. {(4, 1), (5, 2), (8, 2), (9, 8)} Yes, no x-coordinate is repeated.
Write the domain and range in {set notation}
Graphs of a Function Vertical Line Test: If a vertical line is passed over the graph and it intersects the graph in exactly one point, the graph represents a function.
Yes D: all real numbers R: all real numbers Yes D: all real numbers R: y ≥ -6 x x y y Does the graph represent a function? Name the domain and range.
No D: x ≥ 1/2 R: all reals No D: all reals R: all reals x x y y Does the graph represent a function? Name the domain and range.
Yes D: all reals R: y ≥ -6 No D: x = 2 R: all reals x x y y Does the graph represent a function? Name the domain and range.
Function Notation • When we know that a relation is a function, the “y” in the equation can be replaced with f(x). • f(x) is simply a notation to designate a function. It is pronounced ‘f’ of ‘x’. • The ‘f’ names the function, the ‘x’ tells the variable that is being used.
Introducing…… Interval Notation!
Yes D: R: Yes D: R: x x y y List the domain and range in interval notation
No D: R: No D: R: x x y y List the domain and range in interval notation
Yes D: R: No D: R: x x y y List the domain and range in interval notation
List the domain and range in interval notation
Order of Operations(What we use when evaluating the expressions) • P • E • M • D • A • S Please excuse my dear Aunt Sally? *Multiply/Divide and Add/Subtract from left to right!
Examples
Examples • Evaluate the expression.
