GATE 2025 EC GA Q.5: solution and answer

GATE 2025 · ECGA Q.5 · 1 mark · MCQ

Which one of the following plots represents $f (x) = - \frac{∣ x ∣}{x}$ , where $x$ is a non-zero real number?

Note: The figures shown are representative.

GATE 2025 EC GA question 5: Each option is a graph on axes with a horizontal x-axis (arrow to the right, labelled x) and a vertical f(x)-axis (arrow… — Figure from the GATE 2025 EC paper. It is part of the question.

(A)
Figure — horizontal segment at height 1 for x < 0 (ending at (0,1)), horizontal segment at height -1 for x > 0 (starting at (0,-1)).
(B)
Figure — horizontal segment at height 1 for x > 0 (starting at (0,1)), horizontal segment at height -1 for x < 0 (ending at (0,-1)).
(C)
Figure — vertical segment rising from the x-axis at (-1,0), and vertical segment falling from the x-axis at (1,0).
(D)
Figure — two curved branches: for x < 0 the curve hugs the x-axis from above and rises steeply near the vertical axis towards (0,1); for x > 0 the curve rises from far below near (0,-1) and flattens towards the x-axis from below.

Answer and solution

ANSWER

(A). For $x \neq = 0$ , $\frac{∣ x ∣}{x} = sgn (x)$ , so $f (x) = - sgn (x)$ : it equals $+ 1$ for $x < 0$ and $- 1$ for $x > 0$ . That is two horizontal segments, the upper one on the left, as in plot (A).

Official answer: ASource: Official answer key, GATE 2025 EC (IIT Roorkee)

By the definition of absolute value, $∣ x ∣ = x$ when $x > 0$ and $∣ x ∣ = - x$ when $x < 0$ . The function is not defined at $x = 0$ , which is why the question says $x$ is non-zero.
For $x > 0$ : $f (x) = - \frac{x}{x} = - 1$ , a constant. Check: $x = 2$ gives $- \frac{2}{2} = - 1$ .
For $x < 0$ : $f (x) = - \frac{- x}{x} = + 1$ , a constant. Check: $x = - 3$ gives $- \frac{3}{- 3} = + 1$ .
So $f (x) = - sgn (x)$ . Its graph is a horizontal line at height $+ 1$ to the left of the vertical axis and a horizontal line at height $- 1$ to the right of it, with no point at $x = 0$ . The labels $(0, 1)$ and $(0, - 1)$ only mark the open ends of the two segments; the note says the figures are representative.
Now eliminate the other options. (B) has $+ 1$ for $x > 0$ and $- 1$ for $x < 0$ , which is $+ sgn (x)$ , the sign-flipped function. (C) is made of vertical segments at $x = \pm 1$ , which cannot be the graph of any function because one $x$ would have many values. (D) is a pair of curves whose height changes with $x$ and approaches $0$ as $∣ x ∣$ grows, but $f$ is constant on each side.
Only (A) shows $f (x) = 1$ for $x < 0$ and $f (x) = - 1$ for $x > 0$ . The answer is (A).

KEY IDEA

For any non-zero $x$ , $\frac{∣ x ∣}{x}$ is the sign of $x$ : it is $+ 1$ when $x > 0$ and $- 1$ when $x < 0$ . So $f (x) = - \frac{∣ x ∣}{x}$ is a step function. It is constant on each side of the origin, undefined at $x = 0$ , and the leading minus sign puts the $+ 1$ level on the negative side.

WATCH OUT FOR

Dropping the leading minus sign and choosing (B), which is the graph of $\frac{∣ x ∣}{x}$ and not of $- \frac{∣ x ∣}{x}$ . Another error is treating $\frac{∣ x ∣}{x}$ like $\frac{1}{x}$ and choosing the hyperbola-like curves in (D). Test one positive and one negative value of $x$ to rule out both.

From the GATE 2025 EC paper. Every question of this paper · Download the full paper · All 58 Numerical questions, with notes

GATE 2025 EC GA Q.5: solution and answer

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