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
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 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
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
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
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
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
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 a Specific Value Use the graph of to describe its end behavior. Support the conjecture numerically. 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