Let S = \{(x)\}, y): x² + y² = 1, -1 ≤ x { in } R ≤ 1 { and } -1 ≤ y { in } R ≤ 1\} Which one of the following is correct?

Correct Answer is : S is not a function

Correct Answer is : S is not a function

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UPSC NDA II 2018 Mathematics Solved Paper

Question : 25 of 120

Marks: +1, -0

Let

S = \{(x)\}, y): x^{2} + y^{2} = 1, -1 \le x \text{ in } R \le 1 \text{ and } -1 \le y \text{ in } R \le 1\}

Which one of the following is correct?

Solution:

Concept:

A function maps every input to exactly one output. The vertical line test checks this visually.

Explanation:

The set

S = \{(x, y): x^{2} + y^{2} = 1, -1 \leq x \in \mathbb{R} \leq 1, -1 \leq y \in \mathbb{R} \leq 1\}

represents all points on the unit circle.

For any

x

with

-1 < x < 1

, the equation

y = \pm \sqrt{1 - x^{2}}

gives two distinct

y

values.

So a vertical line at such

x

cuts the circle at two points, meaning one input has two outputs.

Hence,

S

fails the vertical line test and is not a function.

Answer:

S

is not a function. Therefore, Option D is correct.

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