Questions: Consider the function g defined by g(x)=c/(x-17) for x<10, C+3x for x=10, 2C-4 for x>10. The symbol C represents the same real number in each of the piecewise formulas. Find the value for C that makes the function continuous on (-∞, 10]. Make sure that your reasoning is clear. Is it possible to find a value for C that makes the function continuous over (-∞, ∞) ? Explain.

Consider the function g defined by g(x)=c/(x-17) for x<10, C+3x for x=10, 2C-4 for x>10. The symbol C represents the same real number in each of the piecewise formulas. Find the value for C that makes the function continuous on (-∞, 10]. Make sure that your reasoning is clear. Is it possible to find a value for C that makes the function continuous over (-∞, ∞) ? Explain.
Transcript text: Consider the function $g$ defined by $g(x)=\left\{\begin{array}{ll}\frac{c}{x-17} & x<10 \\ C+3 x & x=10 \\ 2 C-4 & x>10\end{array}\right.$ The symbol $C$ represents the same real number in each of the piecewise formulas. Find the value for $C$ that makes the function continuous on $(-\infty, 10]$. Make sure that your reasoning is clear. Is it possible to find a value for $C$ that makes the function continuous over $(-\infty, \infty)$ ? Explain.
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Solution

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Solution Steps

To make the function continuous on (,10](- \infty, 10], we need to ensure that the left-hand limit as xx approaches 10 from the left is equal to the value of the function at x=10x = 10. This means solving the equation limx10cx17=C+3×10\lim_{x \to 10^-} \frac{c}{x-17} = C + 3 \times 10.

Step 1: Understanding Continuity at x=10 x = 10

To ensure the function g(x) g(x) is continuous at x=10 x = 10 , the left-hand limit as x x approaches 10 from the left must equal the value of the function at x=10 x = 10 .

The function is defined as:

  • g(x)=cx17 g(x) = \frac{c}{x-17} for x<10 x < 10
  • g(x)=C+3x g(x) = C + 3x for x=10 x = 10

The left-hand limit as x10 x \to 10^- is: limx10g(x)=limx10cx17 \lim_{x \to 10^-} g(x) = \lim_{x \to 10^-} \frac{c}{x-17}

Substituting x=10 x = 10 into the expression: limx10cx17=c1017=c7 \lim_{x \to 10^-} \frac{c}{x-17} = \frac{c}{10-17} = \frac{c}{-7}

The value of the function at x=10 x = 10 is: g(10)=C+3×10=C+30 g(10) = C + 3 \times 10 = C + 30

For continuity at x=10 x = 10 : c7=C+30 \frac{c}{-7} = C + 30

Step 2: Solving for C C

Rearrange the equation for C C : C=c730 C = \frac{c}{-7} - 30

Step 3: Checking Continuity for x>10 x > 10

For the function to be continuous over (,) (-\infty, \infty) , the right-hand limit as x10+ x \to 10^+ must also equal the value of the function at x=10 x = 10 .

The function is defined as:

  • g(x)=2C4 g(x) = 2C - 4 for x>10 x > 10

The right-hand limit as x10+ x \to 10^+ is: limx10+g(x)=2C4 \lim_{x \to 10^+} g(x) = 2C - 4

For continuity at x=10 x = 10 : 2C4=C+30 2C - 4 = C + 30

Step 4: Solving for C C for Continuity Over (,) (-\infty, \infty)

Rearrange the equation: 2CC=30+4 2C - C = 30 + 4 C=34 C = 34

Final Answer

  1. The value for C C that makes the function continuous on (,10] (-\infty, 10] is given by the equation C=c730 C = \frac{c}{-7} - 30 . However, without additional information about c c , we cannot determine a specific value for C C .

  2. It is possible to find a value for C C that makes the function continuous over (,) (-\infty, \infty) . Solving the equation 2C4=C+30 2C - 4 = C + 30 gives C=34 C = 34 .

C=34 \boxed{C = 34}

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