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Warm Up

Represent Relations and Functions. Warm Up. Lesson Presentation. Lesson Quiz. Warm-Up. Evaluate each expression for the given value of x. 1. x 2 + 5 x ; x = –2. –6. ANSWER. 2. 4 x – 3 x 3 ; x = 2. –16. ANSWER. Warm-Up. Evaluate each expression for the given value of x.

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Warm Up

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  1. Represent Relations and Functions Warm Up Lesson Presentation Lesson Quiz

  2. Warm-Up Evaluate each expression for the given value of x. 1.x2 + 5x; x = –2 –6 ANSWER 2. 4x – 3x3; x = 2 –16 ANSWER

  3. Warm-Up Evaluate each expression for the given value of x. 3. –x2 + 3x – 10; x = 3 –10 ANSWER 4. A square flower garden has a perimeter of 24 feet. How long is each side? ANSWER 6 ft

  4. The domain consists of all the x-coordinates:–2, –1, 1, 2,and3. The rangeconsists of all the y-coordinates:–3, –2, 1, and 3. Example 1 Consider the relation given by the ordered pair (–2, –3), (–1, 1), (1, 3), (2, –2), and (3, 1). a.Identify the domain and range. SOLUTION

  5. b. Represent the relation using a graph and a mapping diagram. Example 1 SOLUTION Graph Mapping Diagram

  6. The relation isa function because each input is mapped onto exactly one output. a. Example 2 Tell whether the relation is a function. Explain. SOLUTION

  7. b. The relation isnota function because the input 1 is mapped onto both – 1 and 2. Example 2 Tell whether the relation is a function. Explain. SOLUTION

  8. Guided Practice 1. Consider the relation given by the ordered pairs (–4, 3), (–2, 1), (0, 3), (1, –2), and (–2, –4) a. Identify the domain and range. SOLUTION The domain consists of all the x-coordinates:–4, –2, 0 and 1, The rangeconsists of all the y-coordinates: 3, 1,–2 and –4

  9. Guided Practice b. Represent the relation using a table and a mapping diagram. SOLUTION

  10. 2. Tell whether the relation is a function. Explain. ANSWER Yes; each input has exactly one output. Guided Practice

  11. Example 3 Basketball The first graph below plots average points per game versus age at the end of the 2003–2004 NBA regular season for the 8 members of the Minnesota Timber wolves with the highest averages. The second graph plots average points per game versus age for one team member, Kevin Garnett, over his first 9 seasons. Are the relations shown by the graphs functions? Explain.

  12. Example 3 SOLUTION The team graph does not represent a function because vertical lines at x=28 and x=29 each intersect the graph at more than one point. The graph for Kevin Garnett does represent a function because no vertical line intersects the graph at more than one point.

  13. ANSWER Yes; each input has exactly one output. Example 3 3. WHAT IF? In Example 3, suppose that Kevin Garnett averages 24.2 points per game in his tenth season as he did in his ninth. If the relation given by the second graph is revised to include the tenth season, is the relation still a function? Explain.

  14. Example 4 Graph the equationy= – 2x–1. SOLUTION STEP1 Construct a table of values.

  15. Example 4 STEP 2 Plot the points. Notice that they all lie on a line. STEP3 Connect the points with a line.

  16. a. f (x) = –x2 – 2x + 7 The functionfis not linear because it has an x2-term. Example 5 Tell whether the function is linear.Thenevaluate the function when x= – 4. SOLUTION f (x) =–x2– 2x+ 7 Write function. f (–4) =–(– 4)2– 2(–4) + 7 Substitute–4forx. =–1 Simplify.

  17. b. g(x) = 5x + 8 Example 5 SOLUTION The function gis linear because it has the form g(x) = mx + b. g(x) = 5x+ 8 Write function. Substitute –4 forx. g(–4) = 5(–4) + 8 Simplify. =–12

  18. Guided Practice 4. Graph the equationy = 3x – 2. ANSWER

  19. Guided Practice Tell whether the function is linear. Then evaluate the function when x = –2. 5. f (x) = x – 1 – x3 6. g (x) = –4 – 2x ANSWER ANSWER Not Linear; Whenx = –2,f(x) = 5. Linear; Whenx = –2,f(x) = 0.

  20. Example 6 Diving A diver using a Diver Propulsion Vehicle (DPV) descends to a depth of 130 feet.The pressure P(in atmospheres) on the diver is given by P(d) = 1 + 0.03d where dis the depth (in feet). Graph the function, and determine a reasonable domain and range. What is the pressure on the diver at a depth of 33 feet?

  21. ANSWER At a depth of 33 feet, the pressure on the diver is P(33) = 1 + 0.03(33) 2 atmospheres, which you can verify from the graph. Example 6 SOLUTION The graph ofP(d) is shown.Because the depth varies from0feet to130feet, areasonable domain is0≤d≤130. The minimum value of P(d)isP(0) = 1, and the maximum value of P(d)isP(130) = 4.9. So,a reasonablerange is 1 ≤P(d) ≤4.9.

  22. Guided Practice 7. In 1960, the deep-sea vessel Trieste descended to an estimated depth of 35,800 feet. Determine a reasonable domain and range of the function P(d) in Example 6 for this trip. Because the depth varies from0feet to35,800feet, areasonable domain is0 ≤d≤35,800. The minimum value of P(d)isP(0) = 1, and the maximum value of P(d)isP(35,800) = 1075.So,a reasonablerange is 1≤P(d) ≤1075. SOLUTION

  23. 1. Identify the domain and range of the given relation. Then tell whether the relation is a function. domain: {–3, 0, 5, 8},Range: {–7, 0, 3},function ANSWER Lesson Quiz

  24. ANSWER ANSWER no ; –1 Lesson Quiz 2. Graph y = –2x + 2. 3. Tell whether the function f (x) = –x2 + 3islinear. Then evaluate the function for x =–2.

  25. domain: 1 < c < 10; range: 7 < f (c) < 232 ANSWER Lesson Quiz 4. The average daily income of a physical therapist can be modeled by the function f (c) = 25c – 18, where c is the number of daily customers. Determine a reasonable domain and range for f (c) in this situation.

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