You are asked to determine whether the given function is continuous.

The function is $f(x) = \frac{5}{x + 2}$, and you are asked to determine continuity over the interval (6, 10).

Steps to Check Continuity:

1. Domain of the Function:

   • The function $f(x) = \frac{5}{x + 2}$ is a rational function, and rational functions are continuous everywhere in their domain.
   • The domain is restricted where the denominator is zero. Therefore, we set $x + 2 = 0$, which gives $x = -2$.
   • So, $f(x)$ is undefined at $x = -2$, meaning the function is discontinuous only at $x = -2$.

2. Interval (6, 10):

   • The interval given is (6, 10), which does not include $x = -2$.
   • Therefore, $f(x) = \frac{5}{x + 2}$ is continuous over the interval (6, 10), because it does not contain the point where the function is undefined.

Conclusion:

The function $f(x) = \frac{5}{x+2}$ is continuous on the interval (6, 10).

Would you like more details or have any questions?

Related Questions:

1. What is the domain of a rational function?
2. How do you check for points of discontinuity in a function?
3. What is the formal definition of continuity for a function?
4. How do you determine continuity over an open interval?
5. How do vertical asymptotes affect the continuity of a function?

Tip:

When working with rational functions, always check where the denominator equals zero, as these are the points where the function may be discontinuous.