Chapter 3 functions and graphs 3 1 functions
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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 Graphs 3.1: Functions

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Chapter 3 functions and graphs 3 1 functions

Chapter 3: Functions and Graphs3.1: Functions

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


3 1 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 functions1

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 functions2

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 functions3

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 functions4

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 functions5

3.1: Functions

  • Ex 4: Finding a difference quotient

    • For and h ≠ 0, find each output


3 1 functions6

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 functions7

3.1: Functions

  • Exercises

    • Page 148-149

    • 5-41, odd problems


3 1 functions8

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 functions9

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 functions10

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 functions11

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.


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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 functions13

3.1: Functions

  • Exercises

    • Page 148-149

    • 43-71, odd problems


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