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Key Concept: Types of Discontinuity Concept Summary: Continuity Test Five-Minute Check Then/Now New Vocabulary Key Concept: Limits Key Concept: Types of Discontinuity Concept Summary: Continuity Test Example 1: Identify a Point of Continuity Example 2: Identify a Point of Discontinuity Key Concept: Intermediate Value Theorem Example 3: Approximate Zeros Example 4: Graphs that Approach Infinity Example 5: Graphs that Approach a Specific Value Example 6: Real-World Example: Apply End Behavior Lesson Menu

Use the graph of f (x) to find the domain and range of the function. A. D = , R = B. D = , R = [–5, 5] C. D = (–3, 4) , R = (–5, 5) D. D = [–3, 4], R = [–5, 5] 5–Minute Check 1

Use the graph of f (x) to find the domain and range of the function. A. D = , R = B. D = , R = [–5, 5] C. D = (–3, 4) , R = (–5, 5) D. D = [–3, 4], R = [–5, 5] 5–Minute Check 1

A. y-intercept = 9, zeros: 2 and 3 Use the graph of f (x) to find the y-intercept and zeros. Then find these values algebraically. A. y-intercept = 9, zeros: 2 and 3 B. y-intercept = 8, zeros: 1.5 and 3 C. y-intercept = 9, zeros: 1.5 and 3 D. y-intercept = 8, zero: –1 5–Minute Check 2

A. y-intercept = 9, zeros: 2 and 3 Use the graph of f (x) to find the y-intercept and zeros. Then find these values algebraically. A. y-intercept = 9, zeros: 2 and 3 B. y-intercept = 8, zeros: 1.5 and 3 C. y-intercept = 9, zeros: 1.5 and 3 D. y-intercept = 8, zero: –1 5–Minute Check 2

Use the graph of y = –x 2 to test for symmetry with respect to the x-axis, y-axis, and the origin. A. y-axis B. x-axis C. origin D. x- and y-axis 5–Minute Check 3

Use the graph of y = –x 2 to test for symmetry with respect to the x-axis, y-axis, and the origin. A. y-axis B. x-axis C. origin D. x- and y-axis 5–Minute Check 3

You found domain and range using the graph of a function. (Lesson 1-2) Use limits to determine the continuity of a function, and apply the Intermediate Value Theorem to continuous functions. Use limits to describe end behavior of functions. Then/Now

discontinuous function infinite discontinuity jump discontinuity limit discontinuous function infinite discontinuity jump discontinuity removable discontinuity nonremovable discontinuity end behavior Vocabulary

Key Concept 1

Key Concept 2

Concept Summary 1

Check the three conditions in the continuity test. Identify a Point of Continuity Determine whether is continuous at . Justify using the continuity test. Check the three conditions in the continuity test. 1. Does exist? Because , the function is defined at Example 1

Identify a Point of Continuity 2. Does exist? Construct a table that shows values of f(x) approaching from the left and from the right. The pattern of outputs suggests that as the value of x gets close to from the left and from the right, f(x) gets closer to . So we estimate that . Example 1

Identify a Point of Continuity 3. Does ? Because is estimated to be and we conclude that f (x) is continuous at . The graph of f (x) below supports this conclusion. Example 1

Identify a Point of Continuity Answer: Example 1

Answer: 1. 2. exists. 3. . f (x) is continuous at . Identify a Point of Continuity Answer: 1. 2. exists. 3. . f (x) is continuous at . Example 1

Determine whether the function f (x) = x 2 + 2x – 3 is continuous at x = 1. Justify using the continuity test. A. continuous; f (1) B. Discontinuous; the function is undefined at x = 1 because does not exist. Example 1

Determine whether the function f (x) = x 2 + 2x – 3 is continuous at x = 1. Justify using the continuity test. A. continuous; f (1) B. Discontinuous; the function is undefined at x = 1 because does not exist. Example 1

1. Because is undefined, f (1) does not exist. Identify a Point of Discontinuity A. Determine whether the function is continuous at x = 1. Justify using the continuity test. If discontinuous, identify the type of discontinuity as infinite, jump, or removable. 1. Because is undefined, f (1) does not exist. Example 2

2. Investigate function values close to f(1). Identify a Point of Discontinuity 2. Investigate function values close to f(1). The pattern of outputs suggests that for values of x approaching 1 from the left, f (x) becomes increasingly more negative. For values of x approaching 1 from the right, f (x) becomes increasing more positive. Therefore, does not exist. Example 2

Identify a Point of Discontinuity 3. Because f (x) decreases without bound as x approaches 1 from the left and f (x) increases without bound as x approaches 1 from the right, f (x) has an infinite discontinuity at x = 1. The graph of f (x) supports this conclusion. Answer: Example 2

Answer: f (x) has an infinite discontinuity at x = 1. Identify a Point of Discontinuity 3. Because f (x) decreases without bound as x approaches 1 from the left and f (x) increases without bound as x approaches 1 from the right, f (x) has an infinite discontinuity at x = 1. The graph of f (x) supports this conclusion. Answer: f (x) has an infinite discontinuity at x = 1. Example 2

1. Because is undefined, f (2) does not exist. Identify a Point of Discontinuity B. Determine whether the function is continuous at x = 2. Justify using the continuity test. If discontinuous, identify the type of discontinuity as infinite, jump, or removable. 1. Because is undefined, f (2) does not exist. Therefore f (x) is discontinuous at x = 2. Example 2

2. Investigate function values close to f (2). Identify a Point of Discontinuity 2. Investigate function values close to f (2). The pattern of outputs suggests that f (x) approaches 0.25 as x approaches 2 from each side, so . Example 2

Identify a Point of Discontinuity 3. Because exists, but f (2) is undefined, f (x) has a removable discontinuity at x = 2. The graph of f (x) supports this conclusion. 4 Answer: Example 2

Identify a Point of Discontinuity 3. Because exists, but f (2) is undefined, f (x) has a removable discontinuity at x = 2. The graph of f (x) supports this conclusion. 4 Answer: f (x) is discontinuous at x = 2 with a removable discontinuity. Example 2

A. f (x) is continuous at x = 1. B. infinite discontinuity Determine whether the function is continuous at x = 1. Justify using the continuity test. If discontinuous, identify the type of discontinuity as infinite, jump, or removable. A. f (x) is continuous at x = 1. B. infinite discontinuity C. jump discontinuity D. removable discontinuity Example 2

A. f (x) is continuous at x = 1. B. infinite discontinuity Determine whether the function is continuous at x = 1. Justify using the continuity test. If discontinuous, identify the type of discontinuity as infinite, jump, or removable. A. f (x) is continuous at x = 1. B. infinite discontinuity C. jump discontinuity D. removable discontinuity Example 2

Key Concept 3

Investigate function values on the interval [-2, 2]. Approximate Zeros A. Determine between which consecutive integers the real zeros of are located on the interval [–2, 2]. Investigate function values on the interval [-2, 2]. Example 3

Approximate Zeros Because f (-1) is positive and f (0) is negative, by the Location Principle, f (x) has a zero between -1 and 0. The value of f (x) also changes sign for [1,2]. This indicates the existence of real zeros in each of these intervals. The graph of f (x) supports this conclusion. Answer: Example 3

Approximate Zeros Because f (-1) is positive and f (0) is negative, by the Location Principle, f (x) has a zero between -1 and 0. The value of f (x) also changes sign for [1,2]. This indicates the existence of real zeros in each of these intervals. The graph of f (x) supports this conclusion. Answer: There are two zeros on the interval, –1 < x < 0 and 1 < x < 2. Example 3

Investigate function values on the interval [–2, 2]. Approximate Zeros B. Determine between which consecutive integers the real zeros of f (x) = x 3 + 2x + 5 are located on the interval [–2, 2]. Investigate function values on the interval [–2, 2]. Example 3

Approximate Zeros Because f (-2) is negative and f (–1) is positive, by the Location Principle, f (x) has a zero between –2 and –1. This indicates the existence of a real zero on this interval. The graph of f (x) supports this conclusion. 3 1 –3 Answer: Example 3

Approximate Zeros Because f (-2) is negative and f (–1) is positive, by the Location Principle, f (x) has a zero between –2 and –1. This indicates the existence of a real zero on this interval. The graph of f (x) supports this conclusion. 3 1 –3 Answer: –2 < x < –1. Example 3

B. –3 < x < –2 and –1 < x < 0 A. Determine between which consecutive integers the real zeros of f (x) = x 3 + 2x 2 – x – 1 are located on the interval [–4, 4]. A. –1 < x < 0 B. –3 < x < –2 and –1 < x < 0 C. –3 < x < –2 and 0 < x < 1 D. –3 < x < –2, –1 < x < 0, and 0 < x < 1 Example 3

B. –3 < x < –2 and –1 < x < 0 A. Determine between which consecutive integers the real zeros of f (x) = x 3 + 2x 2 – x – 1 are located on the interval [–4, 4]. A. –1 < x < 0 B. –3 < x < –2 and –1 < x < 0 C. –3 < x < –2 and 0 < x < 1 D. –3 < x < –2, –1 < x < 0, and 0 < x < 1 Example 3

B. Determine between which consecutive integers the real zeros of f (x) = 3x 3 – 2x 2 + 3 are located on the interval [–2, 2]. A. –2 < x < –1 B. –1 < x < 0 C. 0 < x < 1 D. 1 < x < 2 Example 3

B. Determine between which consecutive integers the real zeros of f (x) = 3x 3 – 2x 2 + 3 are located on the interval [–2, 2]. A. –2 < x < –1 B. –1 < x < 0 C. 0 < x < 1 D. 1 < x < 2 Example 3

Graphs that Approach Infinity Use the graph of f(x) = x 3 – x 2 – 4x + 4 to describe its end behavior. Support the conjecture numerically. Example 4

In the graph of f (x), it appears that and Graphs that Approach Infinity Analyze Graphically In the graph of f (x), it appears that and Support Numerically Construct a table of values to investigate function values as |x| increases. That is, investigate the value of f (x) as the value of x becomes greater and greater or more and more negative. Example 4

Graphs that Approach Infinity The pattern of outputs suggests that as x approaches –∞, f (x) approaches –∞ and as x approaches ∞, f (x) approaches ∞. Answer: Example 4

Graphs that Approach Infinity The pattern of outputs suggests that as x approaches –∞, f (x) approaches –∞ and as x approaches ∞, f (x) approaches ∞. Answer: Example 4

Use the graph of f (x) = x 3 + x 2 – 2x + 1 to describe its end behavior. Support the conjecture numerically. A. B. C. D. Example 4

Use the graph of f (x) = x 3 + x 2 – 2x + 1 to describe its end behavior. Support the conjecture numerically. A. B. C. D. Example 4

Graphs that Approach a Specific Value Use the graph of to describe its end behavior. Support the conjecture numerically. Example 5

In the graph of f (x), it appears that . Graphs that Approach a Specific Value Analyze Graphically In the graph of f (x), it appears that . Support Numerically As . As . This supports our conjecture. Example 5

Graphs that Approach a Specific Value Answer: Example 5

Graphs that Approach a Specific Value Answer: Example 5

Use the graph of to describe its end behavior Use the graph of to describe its end behavior. Support the conjecture numerically. A. B. C. D. Example 5

Use the graph of to describe its end behavior Use the graph of to describe its end behavior. Support the conjecture numerically. A. B. C. D. Example 5

Apply End Behavior PHYSICS The symmetric energy function is . If the y-value is held constant, what happens to the value of symmetric energy when the x-value approaches negative infinity? We are asked to describe the end behavior of E (x) for small values of x when y is held constant. That is, we are asked to find . Example 6

Apply End Behavior Because y is a constant value, for decreasing values of x, the fraction will become larger and larger, so . Therefore, as the x-value gets smaller and smaller, the symmetric energy approaches Answer: Example 6

Apply End Behavior Because y is a constant value, for decreasing values of x, the fraction will become larger and larger, so . Therefore, as the x-value gets smaller and smaller, the symmetric energy approaches Answer: Example 6

PHYSICS The illumination E of a light bulb is given by , where I is the intensity and d is the distance in meters to the light bulb. If the intensity of a 100-watt bulb, measured in candelas (cd), is 130 cd, what happens to the value of E when the d-value approaches infinity? A. B. C. D. Example 6

PHYSICS The illumination E of a light bulb is given by , where I is the intensity and d is the distance in meters to the light bulb. If the intensity of a 100-watt bulb, measured in candelas (cd), is 130 cd, what happens to the value of E when the d-value approaches infinity? A. B. C. D. Example 6

End of the Lesson