Chapter 3: Functions and Graphs 3.1: Functions

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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?

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
3.1: 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.

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
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:
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
3.1: Functions
• Ex 4: Finding a difference quotient
• For and h ≠ 0, find each output
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
3.1: Functions
• Exercises
• Page 148-149
• 5-41, odd problems
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.
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
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
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.
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

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