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If f(x)=e^(x) and g(x)=log(e)x, then sho...

If `f(x)=e^(x) and g(x)=log_(e)x,` then show that `"fog=gof"` and find `f^(-1) and g^(-1)`.

Text Solution

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The correct Answer is:
`f^(-1)(x) = log_(e)x, g^(-1)(x) = e^(x)`
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Knowledge Check

  • If f(x)=(1-x)^(1//2) and g(x)=ln(x) then the domain of (gof)(x) is

    A
    `(-oo, 2)`
    B
    `(-1, 1)`
    C
    `(-oo, 1]`
    D
    `(-oo, 1)`

  • Let f(x)=sin x and g(x)= log_(e)|x| . If the ranges of the composition functions fog and gof are R_(1) and R_(2) , respectively, then

    A
    `R_(1)={u: -1 le u lt 1}, { v: -oo v lt 0}`
    B
    `R_(1)={u : - oo lt u lt 0}, R_(2)={v: -oo lt v lt0}`
    C
    `R_(1)={u: -1 lt u lt 1},R_(2)= { v: oo lt v lt 0}`
    D
    `R_(1)={u : -1 le u le 1}, R_(2)={v: -oo lt v lt 0}`

  • If f(x)=|log_e(x||, then f'(x) equals.

    A
    `1/(|x|)`, where `x != 0`
    B
    `1/x"for"|x|gt1and-1/x"for"|x|lt1`
    C
    `-1/x"for"|x|gt1and-1/x"for"|x|lt1`
    D
    `1/x"for"|x|gt0and-1/x"for"|x|lt1`

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    If f(x) = log_(x)(lnx) then f'(x) at x = e is

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