‘The following statement is true. If f(x) sin X, then f is a continuous function; Which of the following is also true? (A) if f(x) is not equal to sin X, then f is not continuous_ (B) if f is not a continuous function; then f(x) is not equal to sin X. (C) if f is continuous, then f(x) sin X’
The Correct Answer and Explanation is:
The correct answer is (B): If f is not a continuous function, then f(x) is not equal to sin x.
Explanation:
The given statement tells us that f(x) = sin x is a continuous function. The continuity of a function means that the function does not have any breaks, jumps, or discontinuities in its domain. Since the sine function, sin x, is continuous everywhere on its domain (which is all real numbers), it follows that if f(x) = sin x, then f(x) is also continuous.
Now, let’s analyze the options:
- Option (A):If f(x) is not equal to sin x, then f is not continuous.
- This statement is false. A function can be different from sin x and still be continuous. For example, f(x) = x^2 is continuous even though it is not equal to sin x.
- Option (B):If f is not a continuous function, then f(x) is not equal to sin x.
- This statement is true. If f(x) is not continuous, it cannot be equal to sin x because sin x is continuous everywhere. Therefore, if a function is not continuous, it cannot be equal to sin x, as the sine function is continuous by definition.
- Option (C):If f is continuous, then f(x) = sin x.
- This statement is false. Just because a function is continuous does not mean it must be equal to sin x. There are many continuous functions that are not equal to sin x, such as f(x) = x^2, f(x) = e^x, etc.
Thus, option (B) is the correct answer.
