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Answers (Lesson 1-3) AA01_A20_PCCRMC01_893802.indd 7 0 1 _ A 2 0 _ P C C R M C Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 0 1 _ 8 9 3 8 0 Chapter 1 2 . i n d d NAME DATE PERIOD NAME DATE PERIOD 7 A7 1-3 Study Guide and Intervention 1-3 Study Guide and Intervention (continued) Continuity, End Behavior, and Limits Continuity, End Behavior, and Limits Continuity A function f(x) is continuous at x = c if it satisfies the End Behavior The end behavior of a function describes how the function behaves at following conditions. either end of the graph, or what happens to the value of f(x) as x increases or decreases without bound. You can use the concept of a limit to describe end behavior. (1) f(x) is defined at c; in other words, f(c) exists. (2) f(x) approaches the same function value to the left and right of c; in other Left-End Behavior (as x becomes more and more negative): lim f(x) x → -∞ words, lim f x exists. ( ) x → c Right-End Behavior (as x becomes more and more positive): lim f(x) (3) The function value that f(x) approaches from each side of c is f(c); in x → ∞ The f(x) values may approach negative infinity, positive infinity, or a specific value. other words, lim f(x) = f(c). x → c Functions that are not continuous are discontinuous. Graphs that are Example Use the graph of f(x) x3 2 to describe y discontinuous can exhibit infinite discontinuity, jump discontinuity, = + 8 its end behavior. Support the conjecture numerically. or removable discontinuity (also called point discontinuity). 4 3 Answers As x decreases without bound, the y-values also f(x) = x + 2 decrease without bound. It appears the limit is negative 4 2 24x Example Determine whether each function is continuous at the given − − 0 4 infinity: lim f(x) = -∞. − x-value. Justify using the continuity test. If discontinuous, identify the type of x → -∞ discontinuity as infinite, jump, or removable. As x increases without bound, the y-values increase −8 2x a. f(x) = 2|x| + 3; x = 2 b. f(x) = − ; x = 1 without bound. It appears the limit is positive infinity: x2 - 1 lim f(x) = ∞. (1) f(2) = 7, so f(2) exists. The function is not defined at x = 1 x → ∞ (Lesson1-3) (2) Construct a table that shows values for because it results in a denominator of 0. Construct a table of values to investigate function values as |x| increases. f(x) for x-values approaching 2 from the The tables show that for values of x left and from the right. approaching 1 from the left, f(x) x -1000 -100 -10 0 10 100 1000 becomes increasingly more negative. For Copyright ©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc. f(x) -999,999,998 -999,998 -998 2 1002 1,000,002 1,000,000,002 x y = f(x) x y = f(x) values approaching 1 from the right, 1.9 6.8 2.1 7.2 f(x) becomes increasingly more positive. 1.99 6.98 2.01 7.02 As x −∞, f(x) -∞. As x ∞, f(x)∞. This supports the conjecture. x y = f(x) x y = f(x) 1.999 6.998 2.001 7.002 0.9 -9.5 1.1 10.5 The tables show that y approaches 7 0.99 -99.5 1.01 100.5 Exercises as x approaches 2 from both sides. 0.999 -999.5 1.001 1000.5 Lesson 1-3 Use the graph of each function to describe its end behavior. Support It appears that lim f(x) = 7. x → 2 The function has infinite discontinuity the conjecture numerically. (3) lim f(x) = 7 and f(2) = 7. at x = 1. y y x → 2 1. 8 2. 8 PDF 2nd The function is continuous at x = 2. 4 4 5x f(x) x 4 2x f(x) = Exercises = - - x - 2 −4 −2 0 24x −16 −8 0 816x Determine whether each function is continuous at the given x-value. −4 −4 Justify your answer using the continuity test. If discontinuous, −8 8 identify the type of discontinuity as infinite, jump, or removable. − Glencoe Precalculus ⎧ 2x + 1 if x > 2 Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. lim f(x)=-∞; lim f(x)=-∞ lim f(x)= 5; lim f(x)= 5 1. f(x) = ; x = 2 2. f(x) = x2 + 5x + 3; x = 4 ( ) ⎨ f 4 = 39 x x x x ⎩ x - 1 if x ≤ 2 → -∞ → ∞ → -∞ → ∞ lim f(x) = 1 and lim f(x) = 5 , lim f(x) = 39 and lim f(x) = 39, See students’ work. See students’ work. x → 2– x → 2+ x → 4- x → 4+ so the function is not continuous; so the function is continuous. it has jump discontinuity. 110/16/09 10:38:02 AM 0 Chapter 1 16 Glencoe Precalculus Chapter 1 17 Glencoe Precalculus / 1 6 / 0 9 1 0 : 3 005_026_PCCRMC01_893802.indd 16 9/30/09 3:02:50 PM 005_026_PCCRMC01_893802.indd 17 3/22/09 5:50:38 PM 8 : 0 2 Answers A M AA01_A20_PCCRMC01_893802.indd 8 0 1 _ A 2 0 _ P C C Chapter 1 R M C 0 1 NAME DATE PERIOD NAME DATE PERIOD _ 8 9 3 8 0 A8 1-3 Practice 1-3 Word Problem Practice 2 . i n d d Continuity, End Behavior, and Limits Continuity, End Behavior, and Limits 8 Determine whether each function is continuous at the given 1. HOUSING According to the U.S. 3. TRIP The per-person cost of a guided x-value(s). Justify using the continuity test. If discontinuous, Census Bureau, the approximate percent climbing expedition can be modeled by 600 identify the type of discontinuity as infinite, jump, or removable. of Americans who owned a home f(x)=− , where x is the number of from 1900 to 2000 can be modeled by x + 25 2 x -2 people on the trip. 1. f(x) = - − ; at x = -1 2. f(x) = − ; at x = -4 h(x)=-0.0009x4 - 0.09x3 + 1.54x2 - 3x2 x + 4 4.12x + 47.37, where x is the number of Yes; the function is defined at No; the function is infinitely a. Graph the function using a graphing decades since 1900. Graph the function x = -1, the function approaches discontinuous at x = -4. calculator. Use the graph to identify 2 on a graphing calculator. Describe the and describe any points of - − as x approaches -1 from end behavior. discontinuity. infinite 3 2 both sides; f(-1) = -− . discontinuity at x = -25 3 lim f(x)=-∞; x x + 1 → -∞ 3. f x x3 2x 2; at x 1 4. f x − ; at x 1 and x 2 b. Are there any points of discontinuity ( ) = - + = ( ) = 2 = - = - x + 3x + 2 lim f(x)=-∞ in the relevant domain? Explain. x → ∞ Answers Yes; the function is defined at No; the function has a removable No, x will not be negative x = -1, the function approaches discontinuity at x = -1 and infinite because the fewest number of 2. GEOMETRY The height of a rectangular 1 as x approaches 1 from discontinuity at x = -2. people is 0. both sides; f(1) = 1. prism with a square base and a volume of 250 cubic units can be modeled by 250 f x − , where x is the length of one ( )= 2 4. STOCK The average price of a share of Determine between which consecutive integers the real zeros of x side of the base. a certain stock x days after a company each function are located on the given interval. (Lesson1-3) restructuring is modeled by a. Determine whether the function is 3 2 3 2 4 f(x)= -0.15x + 1.4x - 1.8x + 15.29. 5. f(x) = x + 5x - 4; [-6, 2] 6. g(x) = x + 10x - 6; [-3, 2] continuous at x = 5. Justify the Stock [-5, -4], [-1, 0], [0, 1] [-3, -2], [0, 1] answer using the continuity test. Copyright ©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc. Yes; because f(5) = 10, the 24 function is defined when Use the graph of each function to describe its end behavior. Support x = 5, lim f(x) = 10. the conjecture numerically. x → 5 12 6 y y b. Is the function continuous? Justify the 7. 8. 8 4 -6x answer using the continuity test. If Price per Share ($) f(x) = 3x 5 246 Lesson 1-3 - 2 0 2 f(x) = x - 4x - 5 4 discontinuous, explain your reasoning and identify the type of discontinuity Day 16 8 816x −8 −4 0 48x − − 0 as infinite, jump, or removable. Use the graph to describe the end 4 −2 − No; because f(0) does not exist, behavior of the function. Support your −4 −8 f(x) is discontinuous at conjecture numerically. x = 0; infinite. lim f(x)=∞; lim f(x)=-∞ lim f(x) = -2; lim f(x) = -2 lim f(x) = ∞; lim f(x) = ∞ c. Graph the function to verify your x → -∞ x → ∞ x → -∞ x → ∞ x → -∞ x → ∞ conclusion from part b. See students’ work. See students’ work. See students’ work. PDF Pass y Glencoe Precalculus Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc. 8 9. ELECTRONICS Ohm’s Law gives the relationship between resistance R, E voltage E, and current I in a circuit as R = − . If the voltage remains 16 8 8 16x I − − 0 constant but the current keeps increasing in the circuit, what happens to −8 the resistance? Resistance decreases and approaches zero.
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