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

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]

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

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

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.

• continuous function

• limit

• discontinuous function

• infinite discontinuity

• jump discontinuity

• removable discontinuity

• nonremovable discontinuity

• end behavior

Identify a Point of Continuity

Check the three conditions in the continuity test.

Determine whether is continuous at

. Justify using the continuity test.

Because , the function is defined at

1. Does exist?

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

.

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.

Identify a Point of Continuity

Answer: 1.

2. exists.

3. .

f (x) is continuous at .

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.

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.

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.

Identify a Point of Discontinuity

Answer: f (x) has an infinite discontinuity at x = 1.

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 infinite discontinuity at x = 1. The graph of f (x) supports this conclusion.

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.

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 .

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.

Answer: f (x) is not continuous at x = 2, with a removable discontinuity.

A. f (x) is continuous at x = 1.

B. infinite discontinuity

C. jump discontinuity

D. removable 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.

Approximate Zeros

Investigate function values on the interval [2, 2].

A. Determine between which consecutive integers

the real zeros of are located on the

interval [–2, 2].

Approximate Zeros

Answer: There are two zeros on the interval, –1 < x < 0 and 1 < x < 2.

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.

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].

Answer: –2 < x < –1.

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 real zeros on this interval. The graph of f (x) supports this conclusion.

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

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

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.

Graphs that Approach Infinity

Analyze Graphically

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.

In the graph of f (x), it appears that and

Graphs that Approach Infinity

The pattern of output suggests that as x approaches –∞, f (x) approaches –∞ and as x approaches ∞, f (x) approaches ∞.

Answer:

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.

Graphs that Approach a Specific Value

Use the graph of to describe its end

behavior. Support the conjecture numerically.

Graphs that Approach a Specific Value

Analyze Graphically

Support Numerically

In the graph of f (x), it appears that

.

As . As . This supports our conjecture.

Graphs that Approach a Specific Value

Answer:

Use the graph of to describe its end

behavior. Support the conjecture numerically.

A.

B.

C.

D.

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 .

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 the value

Answer:

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.

http://connected.mcgraw-hill.com/connected/