Question

Consider the following function and closed interval.

f(x)=x/x-10, [1,9]
Is f continuous on the closed interval [1,9] ?

Answer 1

To determine if f(x) = x/(x-10) is continuous on the closed interval [1,9], we need to check if there are any points of discontinuity within this interval. The function is defined for all x values in the interval [1,9] except x = 10. There are no jumps, holes, or vertical asymptotes within the interval, indicating that the function is continuous.

Step-by-step explanation:

For a function to be continuous on a closed interval, it must be defined and have no jumps, holes, or vertical asymptotes on that interval. To determine if f(x) = x/(x-10) is continuous on the closed interval [1,9], we need to check if there are any points of discontinuity within this interval.

First, check if the function is defined for all x values in the interval. Since the denominator (x-10) would be 0 when x = 10, we need to exclude 10 from the interval. Therefore, the function is defined for all x values in the closed interval [1,9] except x = 10.

Next, check if there are any jumps, holes, or vertical asymptotes. In this case, there are no such discontinuities within the interval [1,9]. Therefore, f(x) = x/(x-10) is continuous on the closed interval [1,9].