Question

Let \( g: \mathbb{R} \to \mathbb{R} \) and \( h: \mathbb{R} \to \mathbb{R} \) be two functions such that \( h(x) = s \cdot g \cdot n(g(x)) \), where \( \mathbb{R} \) denotes the set of all real numbers and \( s \) and \( n \) stand for the signum and the sine functions respectively. Then, select which of the following is not true?

• A. The domain of \( h(x) \) is different from the domain of \( g(x) \) at the same point.
• B. \( h(x) \) is discontinuous at \( g(x) = 0 \).
• C. The domain of \( h(x) \) is the same as the domain of \( g(x) \).
• D. The domain of continuity of \( h(x) \) equals the domain of continuity of \( g(x) - \{ x \in \mathbb{R}, g(x) = 0 \} \).

Hint:

When working with compositions of functions, carefully examine how the domain of each function affects the overall domain, especially when involving piecewise functions like signum.

Answer 1

The Correct Option is A

The function \( h(x) = s \cdot g \cdot n(g(x)) \) involves the composition of several functions. Let's break it down:
- \( g(x) \) is a real-valued function defined for all \( x \in \mathbb{R} \). The domain of \( g(x) \) is the set of real numbers where \( g(x) \) is defined. - \( n(g(x)) \) represents the sine function, which is defined for all real numbers, so it does not introduce any restrictions on the domain. - \( s \) represents the signum function, which is also defined for all real numbers and gives the sign of its argument. Thus, the domain of \( h(x) \) is ultimately determined by the domain of \( g(x) \), since both \( n(g(x)) \) and \( s \) are defined everywhere. The functions involved do not introduce any new domain restrictions. Therefore, the domain of \( h(x) \) is the same as the domain of \( g(x) \), and option (a) is incorrect.
Now, let's analyze each option:
\[\begin{array}{rl} \bullet & \text{(a) The domain of \( h(x) \) is different from the domain of \( g(x) \) at the same point. } \\ \bullet & \text{(b) \( h(x) \) is discontinuous at \( g(x) = 0 \). } \\ \bullet & \text{(c) The domain of \( h(x) \) is the same as the domain of \( g(x) \). } \\ \bullet & \text{(d) The domain of continuity of \( h(x) \) equals the domain of continuity of \( g(x) - \{ x \in \mathbb{R, g(x) = 0 \} \).} } \\ \end{array}\] Thus, the correct answer is (a).