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

2. 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

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

4. Key Concept 1

5. Key Concept 2

6. Concept Summary 1

7. 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

8. 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

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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. Key Concept 3

21. 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

22. 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

23. 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

24. B. Determine between which consecutive integers the real zeros of f (x) = 3x 3 – 2x 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

25. B. Determine between which consecutive integers the real zeros of f (x) = 3x 3 – 2x 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

26. 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

27. 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

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

29. Graphs that Approach a Specific Value
Answer:
Example 5

30. 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

31. 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

32. 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

33. 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