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 $g$.
failed

Solution

failed
failed

Solution Steps

Step 1: Determine the Domain of the Function

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

Step 2: Solve for the Discontinuity

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

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

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

Step 3: Analyze the Interval

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

Final Answer

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

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. \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.}

Was this solution helpful?
failed
Unhelpful
failed
Helpful