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Range of a rational function

This type of question, asked 4 times: May 2021 S1, 16 May 2023 S1, 14 May 2024 S1, 15 May 2026 S1. All 4 questions below, oldest first.

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  1. Question 1

    May 2021 S1

    127
    Range of the function f(x)=x1+x2f(x) = \frac{x}{1+x^2} is

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  2. Question 2

    16 May 2023 S1

    126
    Range of the function f(x)=x2+x+2x2+x+1,  x∈Rf(x) = \frac{x^2+x+2}{x^2+x+1},\; x \in \mathbb{R} is

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  3. Question 3

    14 May 2024 S1

    102
    The range of the function f(x)=x2x2+1f(x)=\dfrac{x^2}{x^2+1} is

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  4. Question 4

    15 May 2026 S1

    51
    Let f:R→Rf : R \rightarrow R be defined by f(x)=x2+2x+3x2+x+1f(x) = \dfrac{x^2 + 2x + 3}{x^2 + x + 1}. If the range of f(x)f(x) is [a,b][a, b], then a+b=…a + b = \ldots

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