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Domain of a composite function fog

This type of question, asked 3 times: 2021 Paper 26, 27 Jan 2024, 23 Jan 2025. All 3 questions below, oldest first.

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  1. Question 1

    2021 Paper 26

    77
    Let f(x)=sin⁡−1xf(x) =\sin^{- 1}x and g(x)=x2−x−22x2−x−6g(x) =\frac{x^{2}-x- 2}{2x^{2}-x- 6}. If g(2)=lim⁡x→2g(x)g(2) =\lim_{x\rightarrow 2} g(x), then the domain of the function fog is:

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  2. Question 2

    27 Jan 2024

    156
    Let f:R−{−12}→Rf:R -\left\{ \frac{- 1}{2} \right\}\rightarrow R and g:R−{−52}→Rg:R -\left\{ \frac{- 5}{2} \right\}\rightarrow R be defined as f(x)=2x+32x+1f(x) =\frac{2x+ 3}{2x+ 1} and g(x)=∣x∣+12x+5g(x) =\frac{|x| + 1}{2x+ 5}. Then the domain of the function fog is:

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  3. Question 3

    23 Jan 2025

    56
    Let f(x)=log⁡exf(x) =\log_{e}x and g(x)=x4−2x3+3x2−2x+22x2−2x+1g(x) =\frac{x^{4}- 2x^{3}+ 3x^{2}- 2x + 2}{2x^{2}- 2x + 1} Then the domain of fog is

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