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forsale [732]
3 years ago
7

Inverse Function QuestionDetermine the expression of f^-1(x) for f(x)=e^x

Mathematics
1 answer:
kifflom [539]3 years ago
6 0

First, find the inverse of f,

y=e^x

x=e^y

Now take the natural logarithm on both sides,

\ln x=\ln e^y\implies f^{-1}(x)=\boxed{\ln(x)}

Second, find the inverse of g,

y=5x\implies g^{-1}(x)=\boxed{\frac{x}{5}}

Now take their composition,

(g\circ f)(x)=g(f(x))=\frac{\ln(x)}{5}

Let y=\frac{\ln(x)}{5}, now again find the inverse,

x=\frac{\ln(y)}{5}

5x=\ln y

exponentiate both sides to base e,

e^{5x}=e^{\ln y}\implies (g\circ f)^{-1}(x)=\boxed{e^{5x}}

Hope this helps :)

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