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Understanding Functions: Domain, Range, and Graphing

Learn how to define domain and range, determine if a relation is a function, and graph equations. Practice evaluating functions and understand key concepts like the Vertical Line Test. Real-world examples included!

dennisreid
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Understanding Functions: Domain, Range, and Graphing

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  1. Splash Screen

  2. Five-Minute Check (over Chapter 1) Then/Now New Vocabulary Key Concept: Functions Example 1: Domain and Range Key Concept: Vertical Line Test Example 2: Real-World Example Example 3: Graph a Relation Example 4: Evaluate a Function Lesson Menu

  3. A B C D A. B. C. D. 5-Minute Check 1

  4. A B C D Name the property illustrated by –15b + 15b = 0. A. Additive Identity B. Multiplicative Inverse C. Inverse Property of Addition D. Symmetry 5-Minute Check 2

  5. A B C D Solve 2(c – 5) – 2 = 8 + c. A. –4 B. 4 C. 10 D. 20 5-Minute Check 3

  6. A B C D A. B. C. D. Solve |3x – 5| + 4 = 14. 5-Minute Check 4

  7. A B C D A. {b | b ≤ 2} B. {b | b < 2} C. {b | b≥ 2} D. {b | b > 2} Solve 2b – 5 ≤ –1. Graph the solution set on a number line. 5-Minute Check 5

  8. A B C D A. eans B. eans C. eans D. eans Which algebraic equation shows the sentence four plus a number divided by six is equal to the product of twelve and the same number? 5-Minute Check 6

  9. You identified domains and ranges for given situations. (Lesson 0–1) • Analyze relations and functions. • Use equations of relations and functions. Then/Now

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

  11. Concept 1

  12. 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, 0), (–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, sothis relation is a function. It is onto, but notone-to-one. Example 1

  13. A B C D 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

  14. Concept 2

  15. 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

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

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

  18. HEALTH The table shows the average weight of a baby for several months during the first year. Graph this information and determine whether it represents a function. Example 2

  19. A B C D A. Yes, this relation is a B. No, this function. relation is not a function. C. Yes, this relation is a D. No, this function. relation is not a function. Example 2

  20. 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

  21. 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 allreal numbers. Example 3

  22. Graph a Relation Determine whether the relation is a function and state whether it is discrete or continuous. Thisgraph 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

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

  24. Evaluate a Function A. Given f(x) = x3 – 3, find f(–2). f(x) = x3 – 3 Original function f(–2) = (–2)3 – 3 Substitute. = –8 – 3 or –11 Simplify. Answer:f(–2) = –11 Example 4A

  25. Evaluate a Function B. Given f(x) = x3 – 3, find f(2t). f(x) = x3 – 3 Original function f(2t) = (2t)3 – 3 Substitute. = 8t3 – 3 (2t)3 = 8t3 Answer:f(2t) = 8t3 – 3 Example 4B

  26. A B C D A. Given f(x) = x2 + 5, find f(–1). A. –4 B. –3 C. 3 D. 6 Example 4A

  27. A B C D B. Given f(x) = x2 + 5, find f(3a). A. 3a2 + 5 B.a2 + 8 C. 6a2 + 5 D. 9a2 + 5 Example 4B

  28. End of the Lesson

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