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Question: How do you find the inverse of \( f\left( x \right) = \dfrac{{{e^x}}}{x} \) ?...

How do you find the inverse of f(x)=exxf\left( x \right) = \dfrac{{{e^x}}}{x} ?

Explanation

Solution

Hint : In the given solution, we have to find the inverse function of the function f(x)=exxf\left( x \right) = \dfrac{{{e^x}}}{x} . Inverse of a function exists only if the given function is onto and one-one. In other words, the function must be a bijective function for its inverse to exist. There is a standard method of finding the inverse function of a given function that we will be using in the given problem.

Complete step-by-step answer :
Now, we have to find the inverse function of f(x)=y=exxf\left( x \right) = y = \dfrac{{{e^x}}}{x} .
So, we have to find the value of x in terms of y.
The given function involves an exponential function expression. So, the inverse of the given function would involve logarithmic function, if it exists.
So, taking natural logarithm on both sides of the equation, we get,
⇒ln⁡y=ln⁡(exx)\Rightarrow \ln y = \ln \left( {\dfrac{{{e^x}}}{x}} \right)
Now, we know a logarithmic property log⁡(ab)=log⁡a−log⁡b\log \left( {\dfrac{a}{b}} \right) = \log a - \log b . So, applying this logarithmic property, we get,
⇒ln⁡y=ln⁡(ex)−ln⁡x\Rightarrow \ln y = \ln \left( {{e^x}} \right) - \ln x
Now, we know a logarithmic property log⁡(xn)=nlog⁡x\log \left( {{x^n}} \right) = n\log x . So, applying this logarithmic property, we get,
⇒ln⁡y=xln⁡(e)−ln⁡x\Rightarrow \ln y = x\ln \left( e \right) - \ln x
Now, we know that the value of the expression ln⁡e\ln e is 11 . So, substituting the value of ln⁡e\ln e in the expression, we get,
⇒ln⁡y=x(1)−ln⁡x\Rightarrow \ln y = x\left( 1 \right) - \ln x
Simplifying further, we get,
⇒ln⁡y=x−ln⁡x\Rightarrow \ln y = x - \ln x
Now, we cannot express x in terms of y further.
So, we should check whether the given function f(x)=exxf\left( x \right) = \dfrac{{{e^x}}}{x} is a bijective function or not.
So, the given function f(x)=exxf\left( x \right) = \dfrac{{{e^x}}}{x} is not a one-one function as there is not a unique image in range for every preimage in the domain of the function.
Hence, the function provided to us is not a bijective function. Therefore, the inverse of function f(x)=exxf\left( x \right) = \dfrac{{{e^x}}}{x} does not exist.

Note : We should first check if the function is a bijective function. If the given function is not a bijective function, then the inverse of the function does not exist. One should be thorough with the method of finding the inverse of different functions in order to solve such questions.