Question

Let f be the function defined by f(x) = x/x for x ≠ 0 and f(0) = ____. Which of the following statements about f is true?

a) f is continuous at x = 0.

b) f is differentiable at x = 0.

c) f is not defined at x = 0.

d) f is increasing on its entire domain.

Answer 1

The function f(x) = x/x for x ≠ 0 and f(0) = ____ is undefined. Thus, option C) f is not defined at x = 0 is the correct choice.

Step-by-step explanation:

The function f(x) = x/x for x ≠ 0 is defined for all real numbers except 0. When x = 0, f(0) = 0/0 which is an indeterminate form in mathematics. Division by zero is undefined in arithmetic, resulting in an indeterminate expression. Therefore, the function is not defined at x = 0, making option C the accurate statement.

The function's discontinuity at x = 0 leads to its undefined nature at that point. Although f(x) approaches 1 as x approaches 0 from either direction, the function does not have a defined value at x = 0. This behavior disrupts the function's continuity and prevents it from being continuous at x = 0.

Understanding the behavior of functions at critical points is essential in calculus and mathematics. Here, the function f(x) = x/x shows an undefined behavior at x = 0 due to division by zero, resulting in an undefined value for f(0). Thus, the function f is not defined at x = 0, leading to the correct choice of option C.