CCSS Content Standards F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Mathematical Practices 1 Make sense of problems and persevere in solving them. 7 Look for and make use of structure.

Then/Now You identified domains and ranges for given situations. Analyze relations and functions. Use equations of relations and functions.

Vocabulary one-to-one function onto function discrete relation continuous relation vertical line test independent variable dependent variable function notation

Concept 1

Example 1 Domain and Range State the domain and range of the relation. Then determine whether the relation is a function. If it is a function, determine if it is one-to-one, onto, both, or neither. The relation is {(1, 2), (3, 3), (0, –2), (–4, –2), (–3, 1)}. Answer:

Example 1 Domain and Range State the domain and range of the relation. Then determine whether the relation is a function. If it is a function, determine if it is one-to-one, onto, both, or neither. The relation is {(1, 2), (3, 3), (0, –2), (–4, –2), (–3, 1)}. Answer: The domain is {–4, –3, 0, 1, 3}. The range is {–2, 1, 2, 3}. Each member of the domain is paired with one member of the range, so this relation is a function. It is onto, but not one-to-one.

Example 1 State the domain and range of the relation shown in the graph. Is the relation a function? A.domain: {–2, –1, 0, 1} range: {–3, 0, 2, 3} Yes, it is a function. B.domain: {–3, 0, 2, 3} range: {–2, –1, 0, 1} Yes, it is a function. C.domain: {–2, –1, 0, 1} range: {–3, 0, 2, 3} No, it is not a function. D.domain: {–3, 0, 2, 3} range: {–2, –1, 0, 1} No, it is not a function.

Example 1 State the domain and range of the relation shown in the graph. Is the relation a function? A.domain: {–2, –1, 0, 1} range: {–3, 0, 2, 3} Yes, it is a function. B.domain: {–3, 0, 2, 3} range: {–2, –1, 0, 1} Yes, it is a function. C.domain: {–2, –1, 0, 1} range: {–3, 0, 2, 3} No, it is not a function. D.domain: {–3, 0, 2, 3} range: {–2, –1, 0, 1} No, it is not a function.

Concept 2

Example 2 TRANSPORTATION The table shows the average fuel efficiency in miles per gallon for SUVs for several years. Graph this information and determine whether it represents a function. Is this relation discrete or continuous?

Example 2 Use the vertical line test. Notice that no vertical line can be drawn that contains more than one of the data points.

Example 2 Answer:

Example 2 Answer: Yes, this relation is a function. Because the graph consists of distinct points, the relation is discrete.

Example 3 Graph a Relation Graph y = 3x – 1 and determine the domain and range. Then determine whether the equation is a function, is one-to-one, onto, both, or neither. State whether it is discrete or continuous. 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.

Example 3 Graph a Relation 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:

Example 3 Graph a Relation 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.

Example 3 Graph a Relation Determine whether the relation is a function and state whether it is discrete or continuous. This graph passes the vertical line test. Every x-value is paired with exactly one unique y-value, and every y-value corresponds to an x-value. Answer:

Example 3 Graph a Relation Determine whether the relation is a function and state whether it is discrete or continuous. This graph passes the vertical line test. Every x-value is paired with exactly one unique y-value, and every y-value corresponds to an x-value. Answer: Yes, the equation y = 3x – 1 represents a function. The function is both one-to-one and onto. Since the domain and range are both all real numbers, the relation is continuous.

Example 3 Graph y = 2x + 5. A.B. C.D.

Example 3 Graph y = 2x + 5. A.B. C.D.

Example 4A Evaluate a Function A. Given f(x) = x 3 – 3, find f(–2). f(x)=x 3 – 3Original function f(–2)=(–2) 3 – 3Substitute. =–8 – 3 or –11Simplify. Answer:

Example 4A Evaluate a Function A. Given f(x) = x 3 – 3, find f(–2). f(x)=x 3 – 3Original function f(–2)=(–2) 3 – 3Substitute. =–8 – 3 or –11Simplify. Answer: f(–2) = –11

Example 4B Evaluate a Function B. Given f(x) = x 3 – 3, find f(2t). Answer: f(x) =x 3 – 3Original function f(2t)=(2t) 3 – 3Substitute. =8t 3 – 3 (2t) 3 = 8t 3

Example 4B Evaluate a Function B. Given f(x) = x 3 – 3, find f(2t). Answer: f(2t) = 8t 3 – 3 f(x) =x 3 – 3Original function f(2t)=(2t) 3 – 3Substitute. =8t 3 – 3 (2t) 3 = 8t 3

Example 4A A.–4 B.–3 C.3 D.6 A. Given f(x) = x 2 + 5, find f(–1).

Example 4A A.–4 B.–3 C.3 D.6 A. Given f(x) = x 2 + 5, find f(–1).

Example 4B B. Given f(x) = x 2 + 5, find f(3a). A.3a B.a C.6a D.9a 2 + 5

Example 4B B. Given f(x) = x 2 + 5, find f(3a). A.3a B.a C.6a D.9a 2 + 5