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If the functions f and g defined from th...

If the functions f and g defined from the set of real number R to R such that `f(x) = e^(x)` and g(x) = 3x - 2, then find functions fog and gof. Also, find the domain of the functions `(fog)^(-1)` and `(gof)^(-1)`.

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The correct Answer is:
A, B, C, D
`(fog)x=f(3x-2)=e^(3x-2)`
and `(gof)x=g(e^(x))=3e^(x)-2`
Let `(fog)x=yimpliese^(3x-2)=y`
`implies 3x-2=log_(e)yimpliesx=(2+log_(e)y)/(3)`
`implies(fog)^(-1)(y)=(2+log_(e)y)/(3)`
implies `y gt 0` So, domain of `(fog)^(-1)` is `(0, oo)`.
Now, again let `(gof) x = 3e^(x) - 2`
`implies y = 3e^(x)-2impliese^(x)=(y+2)/(3)`
`therefore x = log_(e)((y+2)/(3))`
`implies (gof)^(-1)(y)=log_(e)((y+2)/(3))`
Clearly, `y+2gt0impliesygt-2`
`therefore` Domain of `(gof)^(-1)` is `(-2, oo)`.
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