Which of the pairs given above is/are correctly matched?

Question

Consider the following pairs:

a. F = (x,y) ∈ R×R : x² + y = 2

b. F = (x,y) ∈ R×R : x² + y² = 25

c. F = (x,y) ∈ R×R : y = 2x + 1

d. F = (x,y) ∈ R×R : y = x²

Match with:

1. F∘F = (x,z) : z = 4x + 3

2. F increasing on (0,∞), but not on R

3. F is one-one but not onto

4. Range of F = [-2, ∞)

5. F is not a function

Which of the pairs given above is/are correctly matched?

1. A. a-1, b-2, c-3, d-5
2. B. a-2, b-1, c-5, d-4
3. C. a-3, b-4, c-1, d-2
4. D. a-4, b-5, c-1, d-2 (Correct answer)

Correct Answer

Option D — a-4, b-5, c-1, d-2

Detailed Solution & Explanation

The correct answer is a-4, b-5, c-1, d-2.

Key Points

• For F={(x,y):y=2x+1}: composing gives z=4x+3 (option 1) ✓.
• For F={(x,y):x²+y²=25}: circle, fails vertical line test → not a function (option 5) ✓.
• For F={(x,y):y=x²}: increasing on (0,∞) but not all of R (option 2) ✓.
• For F={(x,y):x²+y=2}: y=2−x², range (−∞,2] (option 4).
• Match: a-4, b-5, c-1, d-2.

Additional Information

• A relation is a function only if each x gives exactly one y — the vertical line test. x² + y² = 25 fails it, since most x values give two y values.
• Composition (f ∘ g)(x) = f(g(x)) is applied right to left, and is generally not commutative.
• Functions are one-one (injective) if distinct inputs give distinct outputs, onto (surjective) if every element of the codomain is attained, and bijective if both — only bijections have inverses.
• y = 2x + 1 is linear and bijective on R; x² + y = 2 rearranges to y = 2 − x², a downward parabola and a valid function.