Questions: Is the function given by g(x)=1/(x+8) continuous over the interval (-7,7) ? Why or why not? Select the correct answer below and, if necessary, fill in the answer box to complete your choice. A. No, the function is not continuous at x= . (Use a comma to separate answers as needed.) B. Yes, the function is continuous over (-7,7) because g(x) is a rational function and the values over the interval (-7,7) are in the domain of g.

Is the function given by g(x)=1/(x+8) continuous over the interval (-7,7) ? Why or why not?

Select the correct answer below and, if necessary, fill in the answer box to complete your choice.
A. No, the function is not continuous at x= .
(Use a comma to separate answers as needed.)
B. Yes, the function is continuous over (-7,7) because g(x) is a rational function and the values over the interval (-7,7) are in the domain of g.

Transcript text: Is the function given by $g(x)=\frac{1}{x+8}$ continuous over the interval $(-7,7)$ ? Why or why not? Select the correct answer below and, if necessary, fill in the answer box to complete your choice. A. No, the function is not continuous at $\mathrm{x}=$ $\square$ ]. (Use a comma to separate answers as needed.) B. Yes, the function is continuous over $(-7,7)$ because $g(x)$ is a rational function and the values over the interval $(-7,7)$ are in the domain of $g$.

Solution

Solution Steps

Step 1: Determine the Domain of the Function

The function given is $ g(x) = \frac{1}{x+8} $. This is a rational function, and it is defined for all $ x $ except where the denominator is zero. Therefore, we need to find the values of $ x $ for which $ x+8 = 0 $.

Step 2: Solve for the Discontinuity

Set the denominator equal to zero and solve for $ x $:

\[ x + 8 = 0 \implies x = -8 \]

The function $ g(x) $ is not defined at $ x = -8 $.

Step 3: Analyze the Interval

The interval given is $(-7, 7)$. We need to check if the point of discontinuity $ x = -8 $ lies within this interval. Since $-8$ is not within the interval $(-7, 7)$, the function is continuous over this interval.

Final Answer

The function is continuous over the interval $(-7, 7)$ because the point of discontinuity $ x = -8 $ is not within this interval. Therefore, the correct answer is:

\[ \boxed{\text{B. Yes, the function is continuous over } (-7,7) \text{ because } g(x) \text{ is a rational function and the values over the interval } (-7,7) \text{ are in the domain of } g.} \]

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