Question

Find the domain and range of the function ${e^{\ln x}}$.

Answer 1

Hint: In the given question, we have to find the range of the given composite function in the variable x for all possible real values of x. So, in order to find the domain and range of the function, we will first consider the function as y, and then find the values of x that can be put in the function. Then, we will simplify the function using exponential and logarithmic properties to find the range of function.

Complete step by step solution:
We have the function, ${e^{\ln x}}$.
For the domain, we figure out the values of x that can be put in the function.
We know that only positive values can be put in the logarithmic function. Also, there is no boundation over the values to be put in the exponential function.
The only constraint for the domain of function is $x > 0$.
So, the domain of the function is $\left( {0,\infty } \right)$.
Now, we know the logarithmic property ${x^{\ln y}} = {y^{\ln x}}$. So, we get,
$ \Rightarrow {e^{\ln x}} = {x^{\ln e}}$
Now, we also know that $\ln e = 1$. So, we get,
$ \Rightarrow {e^{\ln x}} = x$
Now, we can see that the set of values of x attained by the function is equal to the values of x that can be put in the function itself.
So, the range of the function is equal to the domain of the function.
Hence, Range of the function is $\left( {0,\infty } \right)$.

Note: Range of a function is the set of values assumed by the function for different values of variables in the domain. Each preimage in the domain has a unique image in the range for a function. So, for each value of variable in function, we get a unique value of function. There are different methods to find the range of different types of functions. The one used here can be used to find the range of any rational function in a variable.