Questions
Single choice
Find the value of c that makes the following function continuous for all real numbers. 𝑓 ( 𝑥 ) = { 𝑐 𝑥 2 + 3 if 𝑥 < − 2 𝑐 𝑥 − 1 if 𝑥 ≥ − 2
Options
A.−
1
3
B.−
2
3
C.1
D.2
3
E.1
3
F.−
1
G.−
2
H.2
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Verified Answer
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Step-by-Step Analysis
We start by identifying the two pieces of the function and the point where continuity must be checked: at x = -2, since the definition changes there.
Left-hand expression (for x < -2): f(x) = c x^2 + 3. As x approaches -2 from the left, the value tends to c(-2)^2 + 3 = 4c + 3.
Right-hand expression (for x ≥ -2): f(x) = c x - 1. At x = -2, the function value is f(-2) = c(-2) - 1 = -2c - 1.
For continuity at x = -2, the left-hand......Login to view full explanationLog in for full answers
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Similar Questions
Determine the following statements are True or False. 1) There is no solution of 𝑒 𝑥 + 𝑒 − 𝑥 2 = 2 on [-2,2]. [ Select ] True False 2) Let f be a continuous function on [0,1] such that 0 < 𝑓 ( 𝑥 ) < 1 for all 𝑥 ∈ [ 0 , 1 ] . We can conclude that there exists a point 𝑎 ∈ [ 0 , 1 ] such that 𝑓 ( 𝑎 ) = 𝑎 . [ Select ] True False 3) If a function f is continuous on [a,b], then there is a c in [a,b] with 𝑓 ( 𝑐 ) = 𝑓 ( 𝑎 ) + 𝑓 ( 𝑏 ) 2 . [ Select ] True False 4) The function 𝑓 ( 𝑥 ) = 1 + 𝑥 2 𝑥 2 − 4 has a maximum on [-3,3]. [ Select ] True False
Find numbers a and b, or k, so that f is continuous at every point.
The function [math: f(x)={−2x+4if x<0,−4x−6if x>0]f(x)=\left \{\begin {array}{ll}-2x+4&\text {if }x<0,\\-4x-6&\text {if }x>0\end {array}\right . is continuous.
Suppose we know the following information about the function 𝑓 ( 𝑥 ) : 𝑓 ( − 1 ) = − 4 , 𝑓 ( 2.5 ) = 3 , 𝑓 ( 𝜋 ) = 2.4 and 𝑓 ( 1 ) does not exist lim 𝑥 ⟶ − 1 − 𝑓 ( 𝑥 ) = − 4 lim 𝑥 ⟶ − 1 + 𝑓 ( 𝑥 ) = − 4 lim 𝑥 ⟶ 2.5 + 𝑓 ( 𝑥 ) = − ∞ lim 𝑥 ⟶ 𝜋 𝑓 ( 𝑥 ) = 0 lim 𝑥 ⟶ 8 − 𝑓 ( 𝑥 ) = 3 lim 𝑥 ⟶ 8 + 𝑓 ( 𝑥 ) = 3.01 What does this information tell us about the continuity of 𝑓 ( 𝑥 ) ? At 𝑥 = − 1 , 𝑓 ( 𝑥 ) is/has a [ Select ] jump discontinuity infinite discontinuity continuous discontinuous, but there is not enough information to tell which type there is not enough information to tell anything removable discontinuity . At 𝑥 = 1 , 𝑓 ( 𝑥 ) is/has a [ Select ] infinite discontinuity continuous there is not enough information to tell anything jump discontinuity removable discontinuity discontinuous, but there is not enough information to tell which type . At 𝑥 = 2.5 , 𝑓 ( 𝑥 ) is/has a [ Select ] continuous discontinuous, but there is not enough information to tell which type removable discontinuity jump discontinuity there is not enough information to tell anything infinite discontinuity . At 𝑥 = 𝜋 , 𝑓 ( 𝑥 ) is/has a [ Select ] there is not enough information to tell anything discontinuous, but there is not enough information to tell which type jump discontinuity continuous infinite discontinuity removable discontinuity . At 𝑥 = 8 , 𝑓 ( 𝑥 ) is/has a [ Select ] jump discontinuity there is not enough information to tell anything continuous removable discontinuity infinite discontinuity discontinuous, but there is not enough information to tell which type .
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