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Chapter 3: Functions and Graphs 3.1: Functions - PowerPoint PPT Presentation

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

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Chapter 3: Functions and Graphs3.1: Functions

Essential Question: How are functions different from relations that are not 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

• 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.

• 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

• 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:

• 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

• Ex 4: Finding a difference quotient

• For and h ≠ 0, find each output

• 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

• Exercises

• Page 148-149

• 5-41, odd problems

• 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.

• 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

• 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

• 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.

• 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

• Exercises

• Page 148-149

• 43-71, odd problems