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
Determine whether
is continuous at
. Justify using the continuity test.
Check the three conditions in the continuity test.
1. Does
Because
exist?
, the function is defined at
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
we conclude that f(x) is continuous at
and
. The
graph of f(x) below supports this conclusion.
Identify a Point of Continuity
Answer: 1.
2.
3.
exists.
.
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
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.
Answer: f(x) has an infinite discontinuity at x = 1.
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.
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
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].
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.
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].
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.
Answer: –2 < x < –1.
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
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.
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
In the graph of f(x), it appears that
.
Support Numerically
As
. As
supports our conjecture.
. This
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
larger, so
will become larger and
. 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.
C.
B.
D.