NIMCET 2025 — Mathematics PYQ

To determine which statement is false, let us analyze how the domain and behavior of the composite function $h(x)=\mathrm{sgn}(g(x))$ are established.

1. Understanding the Signum Function Composition

The signum function is defined for all real numbers:

$$\mathrm{sgn}(y)=\{\begin{array}{llc}1& amp;\text{if }y& gt;0\\ 0& amp;\text{if }y=0& \\ -1& amp;\text{if }y& lt;0\end{array}$$

Because $\mathrm{sgn}(y)$ accepts any real number as its input, the composite function $h(x)=\mathrm{sgn}(g(x))$ is defined for exactly the same values of $x$ for which $g(x)$ is defined.

2. Analysis of the Correct Option (Why Option B is NOT True)

• Statement: The domain of $h(x)$ is different from the domain of $g(x)$ at the same point.

• Reasoning: Since both functions are explicitly given as mappings from real numbers to real numbers ($g:\mathbb{R}\to \mathbb{R}$ and $h:\mathbb{R}\to \mathbb{R}$), their domains are identical ($\mathbb{R}$).

• The input to $h(x)$ is evaluated at any point $x$ where $g(x)$ exists. There is no point $x$ where $g(x)$ yields a real number but $h(x)$ fails to exist, because the signum function is defined for all real numbers.

• Therefore, the domains cannot be different at any point. This makes Option B completely false.

3. Verification of Other Options

• Option A is true: The output of a signum function can only be $-1$, $0$, or $1$. Thus, the range of $h(x)$ must be a subset of $\{-1,0,1\}$.

• Option C is true: If $g(c)\ne 0$ and $g(x)$ is continuous, $g(x)$ maintains a constant sign in a small region around $c$. This ensures $h(x)$ stays constant ($1$ or $-1$) near $c$, making it continuous.

• Option D is true: If $g(x)$ is always positive, $\mathrm{sgn}(g(x))$ will always output $1$.

Correct Answer

The statement which is NOT true is:

The domain of $h(x)$ is different from the domain of $g(x)$ at the same point. (Option A)