Linear Relations and Functions B-1 Relations and Functions

ACT WARM-UP Emily scored 145, 170, and 165 in 3 bowling games. What should she score on her next bowling game if she wants to have an average score of exactly 170 for the 4 games? A) 200 B) 195 C) 185 D) 175 E 165 145+170+165+𝑥 4 =170; Solve: 480 + x = 680; x = 200; Therefore, A) 200 is the score needed on her next bowling game.

Objectives Identify the domain and range of relations and functions Determine whether a relation is a function Find functional values

Essential Question How can you determine whether a relation is a function through Ordered pairs, Mapping diagrams, Tables, Graphs, Words?

A relation is a pairing of input values with output values A relation is a pairing of input values with output values. It can be shown as a set of ordered pairs ( x, y ), where x is an input and y is an output. The set of input values for a relation is called the domain, and the set of output values is called the range. The graph of a relation is the set of points in the coordinate plane corresponding to the ordered pairs in the relation. An x-coordinate is sometimes called an abscissa and a y-coordinate is sometimes called an ordinate.

A function is a special type of relation in which each element of the domain is paired with exactly one element of the range. If any input of a relation has more than one output, the relation is not a function. A mapping shows how each member of the domain is paired with each member of the range. -3 2 1 2 4 -1 1 4 3 5 -3 1 5 1 6 One-to-one function Function, not one-to-one Not a function

The relation is {(1, 2), (3, 3), (0, –2), (–4, 0), (–3, 1)}. State the domain and range of the relation shown in the graph. Is the relation a function? The relation is {(1, 2), (3, 3), (0, –2), (–4, 0), (–3, 1)}. Answer: The domain is {–4, –3, 0, 1, 3}. The range is {–2, 0, 1, 2, 3}. Each member of the domain is paired with exactly one member of the range, so this relation is a function. A function whose graph is made up of unconnected points is called a discrete function. Example 1-1a

State the domain and range of the relation shown in the graph State the domain and range of the relation shown in the graph. Is the relation a function? Answer: The domain is {–3, 0, 2, 3}. The range is {–2, –1, 0, 1}. Yes, the relation is a function. Example 1-1b

Fuel Efficiency (mi/gal) Transportation The table shows the average fuel efficiency in miles per gallon for light trucks for several years. Determine whether it represents a function. Year Fuel Efficiency (mi/gal) 1995 20.5 1996 20.8 1997 20.6 1998 20.9 1999 2000 2001 20.4 Answer: Yes, there is only one fuel efficiency rating (output) for each year (input). Example 1-2a

Health The table shows the average weight of a baby for several months during the first year. Determine whether it represents a function. Age (months) Weight (pounds) 1 12.5 2 16 4 22 6 24 9 25 12 26 Answer: Yes, the table represents a function. There is only one weight for each age. Example 1-2d

Every point on a vertical line has the same x-coordinate, so a vertical line cannot represent a function itself, but can be used to determine whether a relation is a function by using the vertical line test on a graph. If no vertical line intersects a graph in more than one point, the graph represents a function. When two points on the graph of a relation are intersected by a vertical line, this means those two points have the same x value but different y values. That is, one domain value is paired with more than one range value.

y x Graph the relation represented by Make a table of values to find ordered pairs that satisfy the equation. Choose values for x and find the corresponding values for y. Then graph the ordered pairs. (2, 5) 2 1 –1 y x –4 (1, 2) –1 2 (0, –1) 5 (–1, –4) Example 1-3a

Find the domain and range. Since x can be any real number, there is an infinite number of ordered pairs that can be graphed. All of them lie on the line shown. Notice that every real number is the x-coordinate of some point on the line. Also, every real number is the y-coordinate of some point on the line. Answer: The domain and range are both all real numbers. (–1, –4) (0, –1) (1, 2) (2, 5) Example 1-3b

Determine whether the relation is a function. This graph passes the vertical line test. For each x value, there is exactly one y value. (–1, –4) (0, –1) (1, 2) (2, 5) Answer: Yes, the equation represents a function. Example 1-3c

y x Graph the relation represented by Make a table. In this case, it is easier to choose y values and then find the corresponding values for x. Then sketch the graph, connecting the points with a smooth curve. 1 2 y x –1 –2 (5, 2) 5 (2, 1) 2 (1, 0) 1 (2, –1) 2 (5, –2) 5 Example 1-4a

Find the domain and range. Every real number is the y-coordinate of some point on the graph, so the range is all real numbers. But, only real numbers that are greater than or equal to 1 are x-coordinates of points on the graph. (1, 0) (2, –1) (5, –2) (5, 2) (2, 1) Answer: The domain is . The range is all real numbers. Example 1-4b

y x Determine whether the relation is a function. 1 2 –1 –2 5 (1, 0) (2, –1) (5, –2) (5, 2) (2, 1) 1 2 y x –1 –2 5 You can see from the table and the vertical line test that there are two y values for each x value except x = 1. Answer: The equation does not represent a function. Example 1-4c

Example 4d Determine whether the relation is a function. From last name to Social Security number. A last name, such as Smith, from the domain would be associated with many different Social Security numbers. The relation from last name to Social Security number is not a function.

Some sets of ordered pairs can be described by using an equation Some sets of ordered pairs can be described by using an equation. When the set of ordered pairs described by an equation satisfies the definition of a function, the equation can be written in functional notation. When an equation represents a function, the variable, usually x, whose values make up the domain is called the independent variable. The other variable, usually y, is called the dependent variable because its values depend on x. When a function is graphed, the independent variable is graphed on the horizontal axis and the dependent variable is graphed on the vertical axis.

The equation y = 2x + 1 can be written as f (x) = 2x + 1 The equation y = 2x + 1 can be written as f (x) = 2x + 1. The symbol f (x) replaces the y and is read “f of x.” The f is just the name of the function. It is not a variable that is multiplied by x. Suppose you want to find the value in the range that corresponds to the element 4 in the domain of the function. This is written as f (4) and is read “f of 4.” The value f (4) is found by substituting 4 for each x in the equation. Therefore, f (4) = 2 (4) + 1 or 9. Letters other than f can be used to represent a function.

Given , find Original function Substitute. Simplify. Answer: Example 1-5a

Given , find Original function Substitute. Answer: Example 1-5c

Given and find each value. b. c. Answer: 6 Answer: 0.625 Answer: Example 1-5d

Essential Question How can you determine whether a relation is a function through Ordered pairs, Mapping diagrams, Tables, Graphs, Words? All of the x’s are different Each member of the domain is paired with exactly one member of the range. All of the x values are different Passes the vertical line test Each element of the domain is associated with exactly one element in the range.

Math Humor Why did the relation need a math tutor? It failed the vertical line test.