Question

Which of the following functions are continuous on the interval 0a. F(x)=(x-1)/(x²-1)

b. g(x)=(x+1)/(x²+1)
c. H(x)=ln(x²-1)

Answer 1

The function F(x) is not continuous on the interval [0, a]. The function g(x) is continuous on the interval [0, a]. The function H(x) is continuous on the interval [0, a] for x values within the interval (-1, 1).

Step-by-step explanation:

A function is considered continuous on an interval if it is defined and has no abrupt changes or breaks on that interval.

Let's analyze each function to determine if it is continuous on the interval [0, a].

1. F(x) = (x-1)/(x²-1)

• The function is defined for all values of x except x=1, since the denominator becomes 0 at x=1.
• Therefore, F(x) is not continuous at x=1 and is not continuous on the interval [0, a].

g(x) = (x+1)/(x²+1)

• The function is defined for all values of x.
• As there are no breaks or abrupt changes, g(x) is continuous on the interval [0, a].

H(x) = ln(x²-1)

• The function is undefined for x values where x²-1 ≤ 0.
• This occurs when x ≤ -1 or x ≥ 1.
• Therefore, H(x) is not defined for x values outside the interval (-1, 1).
• For x values within the interval (-1, 1), H(x) is continuous on the interval [0, a].

To summarize, function F(x) is not continuous on the interval [0, a], function g(x) is continuous on the interval [0, a], and function H(x) is continuous on the interval [0, a] for x values within the interval (-1, 1).