How to use exponential function?

Question

I know how to use exponential function when required in computer calculator but how does it work? I am still studying and our textbooks are not so detailed which gives us the idea how it works. I am using the function but still anyone can explain me in detail about how does it work?

Let $y = e^x $, after taking value of $x = 1,2,3,4,5,6$. Computer screen displays $e^1, e^2, e^3, e^4, e^5, e^6$. Then what? I don't know how to do it now? Please can anyone explain me? I am good learner can anyone help me?

Main purpose to learn exponential function is I got a question as follows set $A =\{(x,y):y=\exp(x) , x\ \text{belongs to}\ \mathbb{R}\ \text{set}\}$ how to write it in ROSTER form?

Roster form : In this method a set is described by listing elements, seperated by commas, within braces {}. The set of vowels of English Alphabet may be described as {a, e, i, o, u}.

Answer 1

I do not know which calculator or computer you use neither in which environment. It can be sufficiently clever to display $e^6$ because it noticed that the exponent is an integer. But, tell me what happens if you enter $e^{6.0}$ or $e^{6.00001}$.

For the practical calculation of an exponential, let me be very simple and consider that $$e^x=\sum_{i=0}^\infty \frac{x^i}{i!}$$ If you add $30$ terms (using $x=6$), you should arrive to a value of $403.4287934925$ while the exact value is $403.4287934927$

Answer 2

The exponential function can be represented by its Taylor series, $$e^x=\sum_{j=0}^{\infty} \frac {x^j}{j!} = 1+x+\frac {x^2}{2!}+ \cdots $$