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Let f,\ g,\ h be real functions given ...

Let `f,\ g,\ h` be real functions given by `f(x)=sinx` , `g(x)=2x` and `h(x)=cosx` . Prove that `fog=go(fh)dot`

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f, g and h are real functions given by `f(x)=sin x, g(x)=2 x and h(x)=cos x`
` Rightarrow f o g(x)=f[g(x)]`
` =f(2 x) ldots ldots . . . `[ Since, `g(x)=2 x ]`
`=sin 2 x . ldots ldots . . .` [ Since, `f(x)=sin x ]`
` =2 sin x cdot cos x`
` Rightarrow g of (x)=g o[f(x) cdot h(x)]`
` =g(sin x cos x) ldots ldots . .[ Since, f(x)=sin x and h(x)=cos x ]`
` =2 sin x cdot cos x . ldots ldots . . . [ Since, g(x)=2 x ]`
` ...
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Knowledge Check

  • Let f(x)=x|x| , g(x)=sin x and h(x)=go f(x) . Then

    A
    h(x) is not differentiable at x = 0
    B
    h(x) is differentiable at x = 0, but `h'(x)` is not continuous at x = 0
    C
    h'(x) continuous at x = 0 but it is not differentiable at x = 0
    D
    h'(x) is differentiable at x = 0

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