GATE EC 2025 Set 1 — Question 5

The function is given by $f(x) = \frac{|x|}{x}$.We can analyze the function for $x > 0$ and $x < 0$:Case 1: If $x > 0$, then $|x| = x$.
So, $f(x) = \frac{x}{x} = 1$.Case 2: If $x < 0$, then $|x| = -x$.
So, $f(x) = \frac{-x}{x} = -1$.The function is not defined for $x = 0$.Therefore, the plot of $f(x)$ should show:

• A horizontal line at $y = 1$ for all $x > 0$.
• A horizontal line at $y = -1$ for all $x < 0$.

Let's examine the given options:Option (A) shows $f(x) = 1$ for $x > 0$ and $f(x) = -1$ for $x < 0$. This matches our analysis.
Option (B) shows $f(x) = 1$ for $x > 0$ and $f(x) = -1$ for $x < 0$, but the labels for the points are (0,1) and (0,-1) which are on the y-axis, not the x-axis. The graph itself is correct, but the labels are misleading for the points shown on the y-axis. However, the question asks for the plot that represents $f(x)$, and the visual representation of the function's behavior is correct in A and B. Comparing A and B, option A has clearer representation of the function's behavior.
Option (C) shows vertical lines, which is incorrect.
Option (D) shows a continuous curve, which is incorrect.Both A and B visually represent the function correctly. However, option A's labels (0,1) and (0,-1) are points on the y-axis, which are not part of the function's domain. But the lines themselves are correct. Option A is the most standard representation.The final answer is $\boxed{\text{A}}$.