1. Splash Screen

2. 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

3. 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

4. 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

5. 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

6. 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

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

8. Key Concept 1

9. Key Concept 2

10. Concept Summary 1

11. 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

12. 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

13. 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

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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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

23. Key Concept 3

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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

30. 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

31. 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

32. 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

33. 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

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

35. 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

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

37. 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

38. 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

39. 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

40. 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

41. End of the Lesson