ConcepTest • Section 1.3 • Question 1 For problems 1-3, the graph in Figure 1.10 is that of y = f(x). Also, use the graphs (I) – (IV) for the answers. Which could be a graph of cf(x)? ConcepTest • Section 1.3 • Question 1

ConcepTest • Section 1.3 • Answer 1 (III) and (IV). In (III) the graph could be –1/2 f(x) and in (IV) the graph could be 2f(x). COMMENT: You could ask students to verbalize the relationship between f(x) and cf(x) for c > 1, 0 < c < 1, and c < –1. ConcepTest • Section 1.3 • Answer 1

ConcepTest • Section 1.3 • Question 2 For problems 1-3, the graph in Figure 1.10 is that of y = f(x). Also, use the graphs (I) – (IV) for the answers. Which could be a graph of f(x) – k? ConcepTest • Section 1.3 • Question 2

ConcepTest • Section 1.3 • Answer 2 (I) could be f(x) – 1. COMMENT: You could ask students to verbalize the relationship between f(x) and f(x) – k for k > 0 and k < 0. ConcepTest • Section 1.3 • Answer 2

ConcepTest • Section 1.3 • Question 3 For problems 1-3, the graph in Figure 1.10 is that of y = f(x). Also, use the graphs (I) – (IV) for the answers. Which could be a graph of f(x – h)? ConcepTest • Section 1.3 • Question 3

ConcepTest • Section 1.3 • Answer 3 (II) could be a graph of f (x+1). COMMENT: You could ask students to verbalize the relationship between f(x) and f (x – h) for h > 0 and h < 0. ConcepTest • Section 1.3 • Answer 3

ConcepTest • Section 1.3 • Question 4 Which of the following functions is the sum of the functions in Figures 1.11 and 1.12? ConcepTest • Section 1.3 • Question 4

ConcepTest • Section 1.3 • Answer 4 (a). On the interval –1 < x < 0, the sum should be negative and concave up. COMMENT: You could follow up by asking for the graph of the difference of the functions in Figures 1.11 and 1.12 (both differences). ConcepTest • Section 1.3 • Answer 4

ConcepTest • Section 1.3 • Question 5 Given the graph of y = sin x in Figure 1.13, determine which of the graphs are those of sin (2x) and sin (3x)? (I) = sin(2x) and (II) = sin(3x) (I) = sin(2x) and (III) = sin(3x) (II) = sin(2x) and (III) = sin(3x) (II) = sin(2x) and (IV) = sin(3x) (III) = sin(2x) and (IV) = sin(3x) ConcepTest • Section 1.3 • Question 5

ConcepTest • Section 1.3 • Answer 5 (b). Replacing x by 2x means the first positive x-intercept will be at x = π/2, so (I) is that of y = sin(2x). Similarly the first positive zero of y = sin(3x) is at x = π/3, so (III) is that of y = sin(3x). COMMENT: Have students also determine the equation for the graphs labeled (II) and (IV). ConcepTest • Section 1.3 • Answer 5

ConcepTest • Section 1.3 • Question 6

ConcepTest • Section 1.3 • Answer 6 (b). The graph has a y-intercept of 1/2. COMMENT: You could have students tell of properties evident in the graphs of the other choices that conflict with those of y = 3/(2 + 4e-x). ConcepTest • Section 1.3 • Answer 6

ConcepTest • Section 1.3 • Question 7 For Problems 7-13, let f and g have values given in the table. f (g(1)) = ConcepTest • Section 1.3 • Question 7

ConcepTest • Section 1.3 • Answer 7 g(1) = 0, so f (g(1)) = f (0) = –2. COMMENT: You can also consider f (g(x)) for x = –2 and x = 2. ConcepTest • Section 1.3 • Answer 7

ConcepTest • Section 1.3 • Question 8 For Problems 7-13, let f and g have values given in the table. f (g(0)) = ConcepTest • Section 1.3 • Question 8

ConcepTest • Section 1.3 • Answer 8 g(0) = 2, so f (g(0)) = f (2) = –1. COMMENT: You can also consider f (g(x)) for x = –2 and x = 2. ConcepTest • Section 1.3 • Answer 8

ConcepTest • Section 1.3 • Question 9 For Problems 7-13, let f and g have values given in the table. f (g(–1)) = ConcepTest • Section 1.3 • Question 9

ConcepTest • Section 1.3 • Answer 9 g(–1) = 1, so f (g(–1)) = f (1) = 2. COMMENT: You can also consider f (g(x)) for x = –2 and x = 2. ConcepTest • Section 1.3 • Answer 9

ConcepTest • Section 1.3 • Question 10 For Problems 7-13, let f and g have values given in the table. If f (g(x)) = 1, then x = ConcepTest • Section 1.3 • Question 10

ConcepTest • Section 1.3 • Answer 10 f(–2) = 1, so g(2) = –2, so x = 2. COMMENT: You can also consider f (g(x)) = a for a = –2, –1, 2. ConcepTest • Section 1.3 • Answer 10

ConcepTest • Section 1.3 • Question 11 For Problems 7-13, let f and g have values given in the table. If f (g(x)) = 0, then x = ConcepTest • Section 1.3 • Question 11

ConcepTest • Section 1.3 • Answer 11 f(–1) = 0, so g(–2) = –1, so x = –2. COMMENT: You can also consider f (g(x)) = a for a = –2, –1, 2. ConcepTest • Section 1.3 • Answer 11

ConcepTest • Section 1.3 • Question 12 For Problems 7-13, let f and g have values given in the table. If g(f(x)) = 2, then x = ConcepTest • Section 1.3 • Question 12

ConcepTest • Section 1.3 • Answer 12 g(0) = 2, and f(–1) = 0, so x = –1. COMMENT: You can also consider g(f(x)) = a for a = –1, 0, 1. ConcepTest • Section 1.3 • Answer 12

ConcepTest • Section 1.3 • Question 13 For Problems 7-13, let f and g have values given in the table. If g(f(x)) = –2, then x = ConcepTest • Section 1.3 • Question 13

ConcepTest • Section 1.3 • Answer 13 g(2) = –2, and f(1) = 2, so x = 1. COMMENT: You can also consider g(f(x)) = a for a = –1, 0, 1. ConcepTest • Section 1.3 • Answer 13

ConcepTest • Section 1.3 • Question 14 For Problems 14-18, let the graphs of f and g be as shown in Figure 1.14. Estimate the values of the following composite functions to the nearest integer. g(f(0)) ≈ ConcepTest • Section 1.3 • Question 14

ConcepTest • Section 1.3 • Answer 14 f(0) ≈ 0, so g(f(0)) ≈ g(0) ≈ 1. COMMENT: You may want to point out that since f(0) = 0, this composition is similar to composing a function with the identity function. ConcepTest • Section 1.3 • Answer 14

ConcepTest • Section 1.3 • Question 15 For Problems 14-18, let the graphs of f and g be as shown in Figure 1.14. Estimate the values of the following composite functions to the nearest integer. g(f(8)) ≈ ConcepTest • Section 1.3 • Question 15

ConcepTest • Section 1.3 • Answer 15 f(8) ≈ 8, so g(f(8)) ≈ g(8) ≈ 5. COMMENT: You may want to point out that since f(8) = 8, this composition is similar to composing a function with the identity function. ConcepTest • Section 1.3 • Answer 15

ConcepTest • Section 1.3 • Question 16 For Problems 14-18, let the graphs of f and g be as shown in Figure 1.14. Estimate the values of the following composite functions to the nearest integer. g(f(3)) ≈ ConcepTest • Section 1.3 • Question 16

ConcepTest • Section 1.3 • Answer 16 f(3) ≈ 1, so g(f(3)) ≈ g(1) ≈ 2. COMMENT: When you are computing g(f(a)) from the graphs of g and f, it is not always necessary to compute f(a). For example, when the horizontal and vertical scales are the same, you can measure the height of f(a) with a straightedge. This distance placed on the x-axis is the new value from which to measure the height of g. The result will be g(f(a)). ConcepTest • Section 1.3 • Answer 16

ConcepTest • Section 1.3 • Question 17 For Problems 14-18, let the graphs of f and g be as shown in Figure 1.14. Estimate the values of the following composite functions to the nearest integer. f(g(2)) ≈ ConcepTest • Section 1.3 • Question 17

ConcepTest • Section 1.3 • Answer 17 g(2) ≈ 3, so f(g(2)) ≈ f(3) ≈ 1. COMMENT: See the Comment for Problem 16. ConcepTest • Section 1.3 • Answer 17

ConcepTest • Section 1.3 • Question 18 For Problems 14-18, let the graphs of f and g be as shown in Figure 1.14. Estimate the values of the following composite functions to the nearest integer. f(g(–1)) ≈ ConcepTest • Section 1.3 • Question 18

ConcepTest • Section 1.3 • Answer 18 g(–1) ≈ 0, so f(g(–1)) ≈ f(0) ≈ 0. COMMENT: See the Comment for Problem 16. ConcepTest • Section 1.3 • Answer 18

ConcepTest • Section 1.3 • Question 19

ConcepTest • Section 1.3 • Answer 19 (b) Substituting g(x) into f(x) gives COMMENT: Students have trouble simplifying . Next you could have them compute g(f(x)). ConcepTest • Section 1.3 • Answer 19

ConcepTest • Section 1.3 • Question 20 For which values of m, n, and b is f(g(x)) = g(f(x)) if f(x) = x + n and g(x) = mx + b? (a) m = 1, n and b could be any number (b) n = 1, m and b could be any number (c) n = 0, m and b could be any number (d) m = 1, n and b could be any number, or n = 0, m and b could be any number. ConcepTest • Section 1.3 • Question 20

ConcepTest • Section 1.3 • Answer 20 (d). After the composition we have: mx + b + n = mx + mn + b. Thus mn = n which implies m = 1 or n = 0. COMMENT: This provides an opportunity to point out the difference between subtracting the same quantity from both sides (always allowable) and canceling the same term from both sides (sometimes letting you lose information like n = 0). ConcepTest • Section 1.3 • Answer 20

ConcepTest • Section 1.3 • Question 21 Given the graphs of the functions g and f in Figures 1.15 and 1.16, which of the following is a graph of f(g(x))? ConcepTest • Section 1.3 • Question 21

ConcepTest • Section 1.3 • Answer 21 (a). Because f(x) = x, we have f(g(x)) = g(x). COMMENT: Follow-up Question. Which graph represents g(f(x))? ConcepTest • Section 1.3 • Answer 21

ConcepTest • Section 1.3 • Question 22 For Problems 22-23, consider the four graphs. Which of these graphs could represent even functions? ConcepTest • Section 1.3 • Question 22

ConcepTest • Section 1.3 • Answer 22 (III) could be the graph of an even function. COMMENT: You could ask the students to give geometric and analytic definitions here. ConcepTest • Section 1.3 • Answer 22

ConcepTest • Section 1.3 • Question 23 For Problems 22-23, consider the four graphs. Which of these graphs could represent odd functions? ConcepTest • Section 1.3 • Question 23

ConcepTest • Section 1.3 • Answer 23 (II) and (IV) could be graphs of odd functions. COMMENT: You could ask the students to give geometric and analytic definitions here. ConcepTest • Section 1.3 • Answer 23

ConcepTest • Section 1.3 • Question 24 Which of the following could be graphs of functions that have inverses? ConcepTest • Section 1.3 • Question 24

ConcepTest • Section 1.3 • Answer 24 (b) and (c). Because these graphs pass the horizontal line test, they could have inverses. COMMENT: You might redefine the other two functions by limiting their domains so they will have inverses. ConcepTest • Section 1.3 • Answer 24

ConcepTest • Section 1.3 • Question 25 If P = f(t) = 3 + 4t, find f -1(P). ConcepTest • Section 1.3 • Question 25

ConcepTest • Section 1.3 • Answer 25 COMMENT: You could also use f(t) = 3 + 8t3, which uses a bit more algebra. ConcepTest • Section 1.3 • Answer 25

ConcepTest • Section 1.3 • Question 26 Which of the following graphs represents the inverse of the function graphed in Figure 1.17? ConcepTest • Section 1.3 • Question 26

ConcepTest • Section 1.3 • Answer 26 (d). The graph of the inverse is a reflection of the function across the line y = x. COMMENT: You could ask the students why each of the other choices fails to be the inverse. ConcepTest • Section 1.3 • Answer 26