1. Chapter 3: Functions and Graphs 3.1: Functions Essential Question: How are functions different from relations that are not functions?

2. 3.1: Functions A function consists of: ▫A set of inputs, called the domain ▫A rule by which each input determines one and only one output ▫A set of outputs, called the range The phrase “one and only one” means that for each input, the rule of a function determines exactly one output ▫It’s ok for different inputs to produce the same output

3. 3.1: Functions Ex 2: Determine if the relations in the tables below are functions a) b) Inputs11233 Output s 56789 Inputs13579 Output s 55785 Because the input 1 is associated to two different outputs (as is the input 3), this relation is not a function Because each output determines exactly one output, this is a function. The fact that 1, 3 & 9 all output 5 is allowed.

4. 3.1: Functions The value of a function that corresponds to a specific input value, is found by substituting into the function rule and simplifying Ex 3: Find the indicated values of a) b) c)

5. 3.1: Functions Functions defined by equations ▫Equations using two variables can be used to define functions. However, not ever equation in two variables represents a function. ▫ If a number is plugged in for x in this equation, only one value of y is produced, so this equation does define a function. The function rule would be:

6. 3.1: Functions Functions defined by equations ▫ If a number is plugged in for x in this equation, two separate solutions for y are produced, so this equation does not define a function. ▫In short, if y is being taken to an even power (e.g. y 2, y 4, y 6,...) it is not a function. y being taken to an odd power (y 3, y 5, y 7, …) does define a function

7. 3.1: Functions Ex 4: Finding a difference quotient ▫For and h ≠ 0, find each output a) b)

8. 3.1: Functions Ex 4 (continued): Finding a difference quotient ▫For and h ≠ 0, find each output c) ▫If f is a function, the quantity is called the difference quotient of f

9. 3.1: Functions Exercises ▫Page 148-149 ▫5-41, odd problems

10. 3.1: Functions Domains ▫The domain of a function f consists of every real number unless… 1)You’re given a condition telling you otherwise  e.g. x ≠ 2 2)Division by 0 3)The n th root of a negative number (when n is even)  e.g.

11. 3.1: Functions Finding Domains (Ex 6) ▫Find the domain:  When x = 1, the denominator is 0, and the output is undefined. Therefore, the domain of k consists of all real number except 1  Written as x ≠ 1 ▫Find the domain:  Since negative numbers don’t have square roots, we only get a real number for u + 2 > 0 → u > -2  Written as the interval [-2, ∞ ) ▫Real life situations may alter the domain

12. 3.1: Functions Ex 8: Piecewise Functions ▫A piecewise function is a function that is broken up based on conditions ▫Find f(-5)  Because -5 < 4, f(-5) = 2(-5)+3 = -10 + 3 = -7 ▫Find f(8)  Because 8 is between 4 & 10, f(8) = (8) 2 – 1 = 64 – 1 = 63 ▫Find the domain of f  The rule of f covers all numbers < 10, (- ∞,10] ▫Discussion: Collatz sequence

13. 3.1: Functions Greatest Integer Function ▫The greatest integer function is a piecewise- defined function with infinitely many pieces. ▫ What it means is that the greatest integer function rounds down to the nearest integer less than or equal to x. ▫ The calculator has a function [int] which can calculate the greatest integer function.

14. 3.1: Functions Ex 9: Evaluating the Greatest Integer Function ▫Let f(x)=[x]. Evaluate the following. a)f (-4.7) = [-4.7] = b)f (-3) = [-3] = c)f (0) = [0] = d)f (5/4) = [1.25] = e)f ( π ) = [ π ] = -5 -3 0 1 3

15. 3.1: Functions Exercises ▫Page 148-149 ▫43-71, odd problems