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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

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.
3 1 functions12
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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