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Continuity and differentiability of a composite of two piecewise functions

This type of question, asked 3 times: 2021 Paper 5, 26 Jun 2022, 11 Apr 2023. All 3 questions below, oldest first.

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  1. Question 1

    2021 Paper 5

    69
    Let the functions f:R→Rf:\mathbb{R}\rightarrow\mathbb{R} and g:R→Rg:\mathbb{R}\rightarrow\mathbb{R} be defined as f(x)={x+2,x<0x2,x≥0f(x)=\begin{cases} x+2, & x<0 \\ x^2, & x\geq 0 \end{cases} and g(x)={x3,x<13x−2,x≥1g(x)=\begin{cases} x^3, & x<1 \\ 3x-2, & x\geq 1 \end{cases}. Then the number of points in R\mathbb{R} where (f∘g)(x)(f\circ g)(x) is NOT differentiable is equal to:

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  2. Question 2

    26 Jun 2022

    67
    Let f,g:R→Rf,g:R \rightarrow R be two real valued functions defined as f(x)={−∣x+3∣, x<0ex, x≥0f(x) =\left\{ \begin{matrix} - |x + 3| & ,\ x < 0 \\ e^{x} & ,\ x \geq 0 \end{matrix} \right. and g(x)={x2+k1x, x<04x+k2, x≥0g(x) =\left\{ \begin{matrix} x^{2}+k_{1}x & ,\ x < 0 \\ 4x +k_{2} & ,\ x \geq 0 \end{matrix} \right., where k1k_{1} and k2k_{2} are real constants. If (gof) is differentiable at x=0x = 0, then (gof) (−4)+( - 4) + (gof) (4) is equal to:

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  3. Question 3

    11 Apr 2023

    170
    Let ff and gg be two functions defined by f(x)={x+1,x<0∣x−1∣,x≥0f(x) =\left\{ \begin{matrix} x + 1, & x < 0 \\ |x - 1|, & x \geq 0 \end{matrix} \right. and g(x)={x+1,x<01,x≥0g(x) =\left\{ \begin{matrix} x + 1, & x < 0 \\ 1, & x \geq 0 \end{matrix} \right.. Then (gof) (x) is

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