NIMCET 2022 — Mathematics PYQ

1. Definition of Modulus Function:

The absolute value function $|x|$ is defined as:

2. Redefining the Function $f(x)$:

By substituting the definition of $|x|$ into $f(x)=x+|x|$, we get:

3. Checking Continuity at $x=0$:

A function is continuous at a point if the Left-Hand Limit (LHL), Right-Hand Limit (RHL), and the function value are all equal.

• Left-Hand Limit (LHL):

  $$\underset{x\to 0^{-}}{\lim }f(x)=\underset{x\to 0^{-}}{\lim }(0)=0$$

• Right-Hand Limit (RHL):

  $$\underset{x\to 0^{+}}{\lim }f(x)=\underset{x\to 0^{+}}{\lim }(2x)=2(0)=0$$

• Value of the Function:

  $$f(0)=0+|0|=0$$

Since $LHL=RHL=f(0)$, the function is continuous at $x=0$.

4. General Continuity:

• For x > 0, $f(x)=2x$, which is a linear polynomial and continuous everywhere in its domain.

• For x < 0, $f(x)=0$, which is a constant function and continuous everywhere in its domain.

Conclusion:

Since the function is continuous for x < 0, x > 0, and at $x=0$, it is continuous for all real numbers.

Correct Option:

A) $x\in (-\infty ,\infty )$