# How do you evaluate the function f(x)=e^x at the value of x=3.2?

Jul 28, 2017

$f \left(3.2\right) \approx 24.53253$

#### Explanation:

$f \left(x\right) = {e}^{x}$

${e}^{x}$ is a transcendental function meaning that is both irrational and cannot be expressed in terms of a finite sequence of the algebraic operations of addition, multiplication, and root extraction.

Hence, ${e}^{x}$ can never (except in the trivial case $x = 0$) be expressed as a fraction, the root of any polynomial with rational coeffients or the sum of any finite series. Thus, it can only ever be approximated by a number of any base.

Several definitions of ${e}^{x}$ exist. Two of the most well known of these are:

${e}^{x} = {\lim}_{n \to \infty} {\left(1 + \frac{x}{n}\right)}^{n}$ (limit known to exist $\forall x \in \mathbb{R}$)

e^x = sum_(n=0) ^oo (x^n)/(n!) (sum known to converge $\forall x \in \mathbb{R}$)

From the second definition above we can approximate ${e}^{3.2}$ as a decimal as follows:

e^3.2 = 1 + 3.2 + 3.2^2/(2!)+ 3.2^3/(3!) + 3.2^4/(4!) + .......

$\approx 24.53253$