Q. A For each given function f, find ... [FREE SOLUTION]

Chapter 1: Q. A (page 154)

For each given function f, find a real number a that makes f continuous at $x = 0,$ if possible.
$(a) f (x) = \{\begin{cases} 3 x + 1, & if x < 0 \\ 2 x + a, & if x 鈮� 0 \end{cases} (b) f (x) = \{\begin{cases} \frac{a}{x + 2}, & if x < 0 \\ 3, & if x = 0 \\ a x + 1, & if x > 0 \end{cases}$

Short Answer

Expert verified

(a) The function $f (x) = \{\begin{cases} 3 x + 1, & if x < 0 \\ 2 x + a, & if x 鈮� 0 \end{cases}$ is continuous when $a = 1 .$

(b) The function $f (x) = \{\begin{cases} \frac{a}{x + 2}, & if x < 0 \\ 3, & if x = 0 \\ a x + 1, & if x > 0 \end{cases}$ cannotcontinue at any value of a.

Step by step solution

Part (a) Step 1. Given information.

The given function is $f (x) = \{\begin{array}{l} 3 x + 1, & if x < 0 \\ 2 x + a, & if x 鈮� 0 \end{array} .$

Part (a) Step 2. Continuity of the function.

The function is continuous when the left-hand limit, right-hand limit, and $f (0)$ are the same.

$\lim_{x 鈫� 0^{-}} f (x) = f (0) \lim_{x 鈫� 0} (3 x + 1) = 2 (0) + a 3 (0) + 1 = a a = 1$

so the function is continuous when $a = 1 .$

Part (b) Step 1. Given information.

The given function is $f (x) = \{\begin{array}{l} \frac{a}{x + 2}, & if x < 0 \\ 3, & if x = 0 \\ a x + 1, & if x > 0 \end{array} .$

Part (b) Step 2. Continuity of the function.

The function is continuous when the left-hand limit, right-hand limit, and $f (0)$ are the same.

Equate the left-hand limit with $f (0) .$

$\lim_{x 鈫� 0^{-}} f (x) = f (0) \lim_{x 鈫� 0^{-}} \frac{a}{x + 2} = 3 \frac{a}{0 + 2} = 3 a = 6$

Equate the right-hand limit with $f (0) .$

$\lim_{x 鈫� 0^{+}} f (x) = f (0) \lim_{x 鈫� 0^{+}} (a x + 1) = 3 a (0) + 1 = 3 1 = 3$

the equation is absurd, so the function is not continuous at any value ofa.

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91影视

For each given function f, find a real number a that makes f continuous at $x = 0,$ if possible.
$(a) f (x) = \{\begin{cases} 3 x + 1, & if x < 0 \\ 2 x + a, & if x 鈮� 0 \end{cases} (b) f (x) = \{\begin{cases} \frac{a}{x + 2}, & if x < 0 \\ 3, & if x = 0 \\ a x + 1, & if x > 0 \end{cases}$

Short Answer

Step by step solution

Part (a) Step 1. Given information.

Part (a) Step 2. Continuity of the function.

Part (b) Step 1. Given information.

Part (b) Step 2. Continuity of the function.

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91影视

For each given function f, find a real number a that makes f continuous at x=0,if possible.(a)f(x)=3x+1,ifx<02x+a,ifx鈮�0(b)f(x)=ax+2,ifx<03,ifx=0ax+1,ifx>0

Short Answer

Step by step solution

Part (a) Step 1. Given information.

Part (a) Step 2. Continuity of the function.

Part (b) Step 1. Given information.

Part (b) Step 2. Continuity of the function.

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

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Study anywhere. Anytime. Across all devices.

For each given function f, find a real number a that makes f continuous at $x = 0,$ if possible.
$(a) f (x) = \{\begin{cases} 3 x + 1, & if x < 0 \\ 2 x + a, & if x 鈮� 0 \end{cases} (b) f (x) = \{\begin{cases} \frac{a}{x + 2}, & if x < 0 \\ 3, & if x = 0 \\ a x + 1, & if x > 0 \end{cases}$