Chapter 3: Functions and Graphs 3.1: Functions

1. Chapter 3: Functions and Graphs3.1: 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 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

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. y2, y4, y6, ...) it is not a function. y being taken to an odd power (y3, y5, y7, …) does define a function

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

8. 3.1: Functions • Ex 4 (continued): Finding a difference quotient • For and h ≠ 0, find each output • If f is a function, the quantityis 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… • You’re given a condition telling you otherwise • e.g. x ≠ 2 • Division by 0 • The nth 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. • f (-4.7) = [-4.7] = • f (-3) = [-3] = • f (0) = [0] = • f (5/4) = [1.25] = • f (π) = [π] = -5 -3 0 1 3

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