Topic 4 Functions Date: August 21, 2017 Topic: Definition of Function, Domain/Range, End Behavior, Input/Out Values

Functions A function is a relation in which each element of the domain is paired with exactly one element of the range. Another way of saying it is that there is one and only one output (y) with each input (x). f(x) y x

Vertical Line Test (pencil test) If any vertical line passes through more than one point of the graph, then that relation is not a function. Are these functions? NOPE! (NOT A FUNCTION) FUNCTION! FUNCTION!

Vertical Line Test FUNCTION! NO! NO WAY! FUNCTION!

Function Notation Input Name of Function Output

A function is a set of ordered pairs of numbers (x, y) such that no x –values are repeated. What are the domain and range of a function? The Domain is the set of all possible x-values in a function. The Range is the set of all possible y-values in a function.

Definition Example {(3, 6), (2, 8), (5, 3)} Domain All the x-coordinates in the function's ordered pairs Range All the y-coordinates in the function's ordered pairs { 3, 2, 5} { 6, 8, 3}

Interval notation  

Sometimes you will be asked to determine a REASONABLE domain or range

The average daily high temperature for the month of May is represented by the function t = 0.2n + 80 Where n is the date of the month. May has 31 days. What is a reasonable estimate of the domain? Answer: 1 ≤ n ≤ 31 What is a reasonable estimate of the range Answer: See next slide

Our function rule is: t = 0.2n + 80 Our domain is 1 ≤ n ≤ 31 Our smallest possible n is 1 Our largest possible n is 31 To find the range, substitute 1 into the equation and solve. Then substitute 31 into the equation and solve.

Our function rule is: t = 0.2n + 80 Substitute a 1 t = 0.2n + 80 t = 0.2(1) + 80 t = 0.2 + 80 t = 80.2 Substitute a 31 t = 0.2n + 80 t = 0.2(31) + 80 t = 6.2 + 80 t = 86.2 Reasonable range is 80.2 ≤ t ≤ 86.2

Example: Joe has an afterschool job at the local sporting goods store Example: Joe has an afterschool job at the local sporting goods store. He makes $6.50 an hour. He never works more than 20 hours in a week. The equation s(h)=6.5h can be used to model this situation. What is the domain for the equation s(h)? What is the range for the equation s(h)?

End Behavior End Behavior: A function’s end behavior is the behavior of the graph f(x) as x approaches positive infinity or negative infinity.

3(1)2+2 1 5 -2 14 3(-2)2+2 Given f(x) = 3x2 + 2, find: 1) f(1) = 5 = 14 3(-2)2+2 -2 14

Algebraically Defined Function Example: is a function. Substitute 5 for x Substitute x+h for x

Numerically Specified Function Example: Suppose that the function f is specified by the following table. x 1 2 3.7 4 f (x) 3.01 -1.03 2.22 0.01 Then, f (0) is the value of the function when x = 0. From the table, we obtain f (0) = 3.01 Look on the table where x = 0 f (2) = 2.22 Look on the table where x = 2 If f (x) = 1, then what is the value of x?