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Chapter 3: Functions and Graphs 3.1: Functions

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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.

- You’re given a condition telling you otherwise

- The domain of a function f consists of every real number unless…

- 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

- Find 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] =

- Let f(x)=[x]. Evaluate the following.

-5

-3

0

1

3

- Exercises
- Page 148-149
- 43-71, odd problems