Question

How can you find the domain and range of exponential functions $f(x) = 2 - {e^{\dfrac{x}{2}}}$?

Answer 1

Hint: We will first find out the possible values of x we can put in the function, the set of those values of x will be known as the domain of the given function. Now, the possible values of f (x) will be the range of the given function.

Complete step-by-step solution:
We are given that we are required to find the domain and range of exponential functions $f(x) = 2 - {e^{\dfrac{x}{2}}}$.
Now, we know that in exponential functions, power can be anything and can take any value.
Therefore, $\dfrac{x}{2} \in \mathbb{R}$, which will then imply that $x \in \mathbb{R}$.
Thus, its domain is whole of real numbers.
Hence, Domain = $\mathbb{R}$.
Now, since we know that ${e^u} > 0$ for all $u \in \mathbb{R}$.
Therefore, we have ${e^{\dfrac{x}{2}}} > 0$
Multiplying by – 1 on both the sides of above equation, we will then obtain the following equation with us:-
$ \Rightarrow - {e^{\dfrac{x}{2}}} < 0$
Adding 2 on both the sides of the above mentioned equation, we will then obtain the following equation with us:-
$ \Rightarrow 2 - {e^{\dfrac{x}{2}}} < 0 + 2$
Simplifying the right hand side of the above equation, we will then obtain the following equation with us:-
$ \Rightarrow 2 - {e^{\dfrac{x}{2}}} < 2$
Thus, the range of the given function is $\left( { - \infty ,2} \right)$.
Hence, the final answer is as follows: Range = $\left( { - \infty ,2} \right)$ and Domain = $\mathbb{R}$.

Note: The students must know the definitions of both the Domain and Range before pursuing any question to find them. If we need to find the domain and range of a function f (x), the domain is the set of possible values of x which can be put in the function and the possible values of f (x) which comes out will be the range of the function.
The students must also note that we have a strictly less than sign in the solution of the above question, therefore, we used the open bracket instead of a closed one. And, thus we have the open brackets on both the sides of the interval $\left( { - \infty ,2} \right)$.