Greatest Integer Function PYQ | JEE Main 2025 | Graph-Based Question

 

❓ Question

Let

$$y=[x]$$

be the Greatest Integer Function (GIF), and

$$y=\{x\}$$

be the Fractional Part Function.

The graph of $y=[x]$ will be parallel to which graph?

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✍️ Solution

Recall the definitions:

$$[x]=\text{Greatest Integer less than or equal to }x,$$
$$\{x\}=x-[x].$$

For example,

$$[5.2]=5,\qquad \{5.2\}=0.2,$$
$$[-5.2]=-6,\qquad \{-5.2\}=0.8.$$

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Graph of $y=[x]$

The graph consists of horizontal line segments.

On each interval

$$n\le x<n+1,$$

we have

$$y=n,$$

which is a horizontal line.

Hence, every step is parallel to the x-axis.

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Graph of $y=\{x\}$

On each interval

$$n\le x<n+1,$$
$$\{x\}=x-n,$$

which is a straight line of slope $1$.

Therefore, each segment is parallel to the line

$$y=x.$$

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✅ Final Answer

$$\boxed{\text{The graph of }y=[x]\text{ is parallel to the x-axis.}}$$

(Also, $y=\{x\}$ is parallel to the line $\boxed{y=x}$.)