154. Consider the graph of the function y = f(x) shown in the following graph. -1 0 1+ 2 a. Find all values for which the function is discontinuous. b. For each value in part a., state why the formal definition of continuity does not apply. c. Classify each discontinuity as either jump, removable, or infinite. [3x, x> 1 1,3 155. Let f(x) = { 1.
154. Consider the graph of the function y = f(x) shown in the following graph. -1 0 1+ 2 a. Find all values for which the function is discontinuous. b. For each value in part a., state why the formal definition of continuity does not apply. c. Classify each discontinuity as either jump, removable, or infinite. [3x, x> 1 1,3 155. Let f(x) = { 1.
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
Related questions
Question
100%
![154. Consider the graph of the function y = f(x) shown
in the following graph.
158. Sketch the graph of the function y= f(x) with
%3D
properties i. through iv.
i. The domain of fis (0, 5]
lim f(x) and lim f(x) exist and are equal.
ii.
iii. f(x) is left continuous but not continuous at
x = 2, and right continuous but not continuous at
x = 3.
iv. f(x) has a removable discontinuity at x= 1, a
1
2
-1t
jump discontinuity at x= 2, and the following
lim_ f(x) = -00
limits
hold:
and
a. Find all values for which the function is
lim f(x) = 2.
discontinuous.
b. For each value in part a., state why the formal
definition of continuity does not apply.
c. Classify each discontinuity as either jump,
removable, or infinite.
In the following exercises, suppose y = f(x) is defined for
all x. For each description, sketch a graph with the indicated
property.
S3x, x> 1
159. Discontinuous at x = 1 with lim f(x) = -1 and
lim f(x) = 4
155. Let f(x) =
a. Sketch the graph of f.
b. Is it possible to find a value k such that f(1) = k,
160. Discontinuous at x= 2 but continuous elsewhere
which makes f(x) continuous for all real
numbers? Briefly explain.
with lim f(x) = }
156. Let f(x)
for x + - 1, 1.
Determine whether each of the given statements is true.
Justify your response with an explanation or
counterexample.
a. Sketch the graph of f.
b. Is it possible to find values k and k2 such that
f(-1) = k, and f(1) = k2, and that makes
161. f(1) = -
is continuous everywhere.
%3D
el -
f(x) continuous for all real numbers? Briefly
162. If the left- and right-hand limits of f(x) as x→ a
explain.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb88ebeb6-9efb-4946-ba9d-4a5fa1ca266a%2Fb35ff663-0c24-4a9c-8af2-63f07dbd6a57%2Ftgl90a_processed.jpeg&w=3840&q=75)
Transcribed Image Text:154. Consider the graph of the function y = f(x) shown
in the following graph.
158. Sketch the graph of the function y= f(x) with
%3D
properties i. through iv.
i. The domain of fis (0, 5]
lim f(x) and lim f(x) exist and are equal.
ii.
iii. f(x) is left continuous but not continuous at
x = 2, and right continuous but not continuous at
x = 3.
iv. f(x) has a removable discontinuity at x= 1, a
1
2
-1t
jump discontinuity at x= 2, and the following
lim_ f(x) = -00
limits
hold:
and
a. Find all values for which the function is
lim f(x) = 2.
discontinuous.
b. For each value in part a., state why the formal
definition of continuity does not apply.
c. Classify each discontinuity as either jump,
removable, or infinite.
In the following exercises, suppose y = f(x) is defined for
all x. For each description, sketch a graph with the indicated
property.
S3x, x> 1
159. Discontinuous at x = 1 with lim f(x) = -1 and
lim f(x) = 4
155. Let f(x) =
a. Sketch the graph of f.
b. Is it possible to find a value k such that f(1) = k,
160. Discontinuous at x= 2 but continuous elsewhere
which makes f(x) continuous for all real
numbers? Briefly explain.
with lim f(x) = }
156. Let f(x)
for x + - 1, 1.
Determine whether each of the given statements is true.
Justify your response with an explanation or
counterexample.
a. Sketch the graph of f.
b. Is it possible to find values k and k2 such that
f(-1) = k, and f(1) = k2, and that makes
161. f(1) = -
is continuous everywhere.
%3D
el -
f(x) continuous for all real numbers? Briefly
162. If the left- and right-hand limits of f(x) as x→ a
explain.
![For the following exercises, determine the point(s), if any,
at which each function is discontinuous. Classify any
discontinuity as jump, removable, infinite, or other.
sino, 0<0<5
cos (0+ k). <0s I
146. f(0) = •
131. f(x) = *
147. f1) = *+3r+2,
k, x= -2
x+ - 2
147. f(x) = {
132. f(x) =
x +1
133. f(x) =
148. f(x) = { e, 0sx<4
u+3, 4sx58
x-x
149. f(x) ={ Vkx, 0sxS3
r+1, 3<xs 10
134. g(1) =1+ I
135. f(x) =
In the following exercises, use the Intermediate Value
Theorem (IVT).
136. f(x) =
- 21
X- 2
150. Let h(x) = 3x - 4, x<2 Over the interval
15+4.x, x> 2
137. H(x) = tan2x
[0, 4], there is no value of x such that h(x) = 10,
although h(0) < 10 and h(4) > 10. Explain why this
138. f(1) = +3
*+ 5t +6
does not contradict the IVT.
151. A particle moving along a line has at each time t
a position function s(1), which is continuous. Assume
For the following exercises, decide if the function
continuous at the given point. If it is discontinuous, what
type of discontinuity is it?
s(2) = 5 and s(5) = 2. Another particle moves such that
its position is given by h(1) = s(1) – 1. Explain why there
139. f(x)2x² - 5x +3 at x= 1
X-1
must be a value c for 2<c < 5 such that h(c) = 0.
152. [T] Use the statement "The cosine of t is equal to t
cubed."
a. Write a mathematical equation of the statement.
b. Prove that the equation in part a. has at least one
real solution.
c. Use a calculator to find an interval of length 0.01
that contains a solution.
140.) h(0) =
sin0 -
cos
at 0 =
tane
2u – 1
if u =
141. g(u) = {
153. Apply the IVT to determine whether 2* = x has
a solution in one of the intervals [1.25, 1.375] or
[1.375, 1.5). Briefly explain your response for each
sin(xy)
tan(æy)'
14
f(y) =
at y = 1
143. f(x) = {r² -eť if x < 0
lr-1 if x 20'
at x= 0
interval.
Jxsin(x) if x Sa
lxtan(x) if x > x'
144. f(x) =
at x =A](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb88ebeb6-9efb-4946-ba9d-4a5fa1ca266a%2Fb35ff663-0c24-4a9c-8af2-63f07dbd6a57%2Fn7p8ae_processed.jpeg&w=3840&q=75)
Transcribed Image Text:For the following exercises, determine the point(s), if any,
at which each function is discontinuous. Classify any
discontinuity as jump, removable, infinite, or other.
sino, 0<0<5
cos (0+ k). <0s I
146. f(0) = •
131. f(x) = *
147. f1) = *+3r+2,
k, x= -2
x+ - 2
147. f(x) = {
132. f(x) =
x +1
133. f(x) =
148. f(x) = { e, 0sx<4
u+3, 4sx58
x-x
149. f(x) ={ Vkx, 0sxS3
r+1, 3<xs 10
134. g(1) =1+ I
135. f(x) =
In the following exercises, use the Intermediate Value
Theorem (IVT).
136. f(x) =
- 21
X- 2
150. Let h(x) = 3x - 4, x<2 Over the interval
15+4.x, x> 2
137. H(x) = tan2x
[0, 4], there is no value of x such that h(x) = 10,
although h(0) < 10 and h(4) > 10. Explain why this
138. f(1) = +3
*+ 5t +6
does not contradict the IVT.
151. A particle moving along a line has at each time t
a position function s(1), which is continuous. Assume
For the following exercises, decide if the function
continuous at the given point. If it is discontinuous, what
type of discontinuity is it?
s(2) = 5 and s(5) = 2. Another particle moves such that
its position is given by h(1) = s(1) – 1. Explain why there
139. f(x)2x² - 5x +3 at x= 1
X-1
must be a value c for 2<c < 5 such that h(c) = 0.
152. [T] Use the statement "The cosine of t is equal to t
cubed."
a. Write a mathematical equation of the statement.
b. Prove that the equation in part a. has at least one
real solution.
c. Use a calculator to find an interval of length 0.01
that contains a solution.
140.) h(0) =
sin0 -
cos
at 0 =
tane
2u – 1
if u =
141. g(u) = {
153. Apply the IVT to determine whether 2* = x has
a solution in one of the intervals [1.25, 1.375] or
[1.375, 1.5). Briefly explain your response for each
sin(xy)
tan(æy)'
14
f(y) =
at y = 1
143. f(x) = {r² -eť if x < 0
lr-1 if x 20'
at x= 0
interval.
Jxsin(x) if x Sa
lxtan(x) if x > x'
144. f(x) =
at x =A
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