Class 12 Relations And Functions · Very Long Answer5 Markstypes_of_functions

Class 12 Maths · Relations And Functions · Very Long Answer #221

For f:R→R, f(x)=x/(x²+1), test one-one and onto and find its range.

✅ Step-by-Step Solution

Final Answer: Neither one-one nor onto R; range [−1/2,1/2].

The denominator x²+1 is positive for every real input.
The distinct inputs 2 and 1/2 both have image 2/5, so f is not injective.
From (x−1)²≥0 and (x+1)²≥0, obtain −(x²+1)≤2x≤x²+1, giving −1/2≤f(x)≤1/2.
The endpoints occur at x=−1 and x=1. Since f is continuous on [−1,1], the intermediate value theorem gives every value between them.
Thus the range is [−1/2,1/2], so f is not onto R and is neither one-one nor onto.

[1 mark] The denominator x²+1 is positive for every real input.
[1 mark] The distinct inputs 2 and 1/2 both have image 2/5, so f is not injective.
[1 mark] From (x−1)²≥0 and (x+1)²≥0, obtain −(x²+1)≤2x≤x²+1, giving −1/2≤f(x)≤1/2.
[1 mark] The endpoints occur at x=−1 and x=1. Since f is continuous on [−1,1], the intermediate value theorem gives every value between them.
[1 mark] Thus the range is [−1/2,1/2], so f is not onto R and is neither one-one nor onto.

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