4.6: Formalizing Relations and Functions

Objective

To determine whether a relation is a function.

Objective To determine whether a relation is a function. To find domain and range using function notation.

Vocab (paragraph): page 268 A relation is a pairing of numbers in one set, called the domain, with numbers in another set, called the range.

Vocab (paragraph): page 268 A relation is often represented as a set of ordered pairs (x, y). In this case, the domain is the set of x-values and the range is the set of y-values.

Essential Understanding A function is a special type of relation in which each value in the domain is paired with exactly one value in the range.

Essential Understanding In short, there can’t be 2 y’s for the same x.

Problem 1 page 268

The diagram they use is completely optional;

Problem 1 page 268 The diagram they use is completely optional; that being said, it may be quite helpful for you to sort out your information.

Got it on the top of page 269 a. (4.2, 1.5), (5, 2.2), (7, 4.8), (4.2, 0)

Got it on the top of page 269 a. (4.2, 1.5), (5, 2.2), (7, 4.8), (4.2, 0) DomainRange

Got it on the top of page 269 a. (4.2, 1.5), (5, 2.2), (7, 4.8), (4.2, 0) DomainRange

Got it on the top of page 269 a. (4.2, 1.5), (5, 2.2), (7, 4.8), (4.2, 0) DomainRange Since 4.2 goes to 2 different y values, this is not a function.

Got it on the top of page 269 b. (-1, 1), (-2, 2), (4, -4), (7, -7) Dom ain Ran ge Since every x goes to one y, this is an example of a function..

2nd way of determining:

Called the vertical line test.

2 nd way of determining: Called the vertical line test. Basically, after graphing the function, if you can draw a vertical line through 2 different points on the graph, it is not a function.

Example: Keep in mind that the square root of a number can be positive or negative.

Example: Keep in mind that the square root of a number can be positive or negative. For example, the square root of 4 is both 2 and -2

Example: Keep in mind that the square root of a number can be positive or negative. For example, the square root of 4 is both 2 and -2 Since x (4) can be mapped to both 2 and -2, this is not a function

Example: Keep in mind that the square root of a number can be positive or negative. For example, the square root of 4 is both 2 and -2 Since x (4) can be mapped to both 2 and -2, this is not a function Validated by the vertical line test.

2 nd part of formalizing:

Notation

2 nd part of formalizing: Basically, we are replacing the dependent variable

2 nd part of formalizing: Basically, we are replacing the dependent variable (often y)

2 nd part of formalizing: Basically, we are replacing the dependent variable (often y) with the notation f(x).

2 nd part of formalizing: Basically, we are replacing the dependent variable (often y) with the notation f(x). y = mx + b

2 nd part of formalizing: Basically, we are replacing the dependent variable (often y) with the notation f(x). f(x) = mx + b

Evaluating functions…

Given the function f(x), replace x with the value assigned and compute arithmetically.

Problem 3 on the bottom of page 269

w(x) = 250x Represents the words you can read in 1 minute

Problem 3 on the bottom of page 269 w(x) = 250x If they ask how many words you can read in 8 minutes, they’re saying…

w(x) = 250x If they ask how many words you can read in 8 minutes, they’re saying… x = 8

w(x) = 250x Replace x with 8 x = 8

w(8) = 250(8) Replace x with 8 x = 8

which is 2000 w(8) = 250(8) Replace x with 8 x = 8

Finding the range of a function given f(x) notation

Given all the values of the domain.

Finding the range of a function given f(x) notation Given all the values of the domain. (1)Plug in each value of the domain into the function expression.

Finding the range of a function given f(x) notation Given all the values of the domain. (2)Evaluate the expression.

Finding the range of a function given f(x) notation Given all the values of the domain. (3)List the results of this as your range.

Example: The domain of f(x) = -1.5x + 4 is {1, 2, 3, 4}. What is the range?

Example: The domain of f(x) = -1.5x + 4 is {1, 2, 3, 4}. What is the range? DomainF(x)Range

Example: The domain of f(x) = -1.5x + 4 is {1, 2, 3, 4}. What is the range? DomainF(x)Range 1-1.5(1)

Example: The domain of f(x) = -1.5x + 4 is {1, 2, 3, 4}. What is the range? DomainF(x)Range 1-1.5(1)

Example: The domain of f(x) = -1.5x + 4 is {1, 2, 3, 4}. What is the range? DomainF(x)Range 1-1.5(1) (2) (3) (4) + 4

Example: The domain of f(x) = -1.5x + 4 is {1, 2, 3, 4}. What is the range? DomainF(x)Range 1-1.5(1) (2) (3) (4) + 4-2

Example: Thus, the range is {-2, -0.5, 1, 2.5} DomainF(x)Range 1-1.5(1) (2) (3) (4) + 4-2

Quickly do the got it underneath.

{-8, 0, 8, 16}

Problem 5 to finish…