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Question #40340

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \{ (x,y) \in \mathbb{R}^2 \mid y - x \le 1, \; x + y < 5, \; x > 0, \; y \ge 0 \}$ . Let $f: S \to \mathbb{R}$ be a function defined by $f(x,y) = x - y$ . If $E$ represents the range of $f$ on the domain $S$ , then $E$ is equal to:

Options

A.
(-1, 5)
B.
[-1, 5)
C.
[-1, 5]
D.
(-1, 5]

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Codomain vs Range Distinction|Boundary Value Theorem|Corner Point Method Precision

Why This Question Is Tricky

Common Mistakes

This question tests the precision of the Corner Point Method in Linear Programming when dealing with strict inequalities and open boundaries.

A standard heuristic is that extrema occur at the vertices.

Since all four vertices of the feasible region are strictly excluded by the boundary conditions ( $x>0$ and $x+y<5$ ), students commonly fall into the trap of assuming that neither the supremum nor the infimum is achieved, incorrectly concluding the range is the open interval $(-1, 5)$ .

However, the objective function is parallel to one of the inclusive boundary edges ( $y-x \le 1$ ).

Consequently, the minimum value is achieved along the entire interior of that edge, which lies perfectly within the feasible region.

This elegantly distinguishes between a limit point and an achieved minimum, rewarding topological rigor.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 11
Concept	Codomain vs Range Distinction
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free