# What is the domain and range of  y=2e^(-x)?

Jul 1, 2017

Domain: $\mathbb{R}$
Range: ${\mathbb{R}}^{+}$

#### Explanation:

My advice is to always graph a function so you can see what it looks like:

graph{2e^-x [-10, 10, -5, 5]}

Just from looking:

Domain: $\mathbb{R}$
Range: ${\mathbb{R}}^{+}$

Does the graph touch zero? How high does it go? Let's be a bit more rigorous and find out by looking at the behaviour as $x$ tends to $\pm \infty$:

${\lim}_{x \to \infty} \left(2 {e}^{-} x\right) = 2 {e}^{- \infty} = \frac{2}{e} ^ \infty = \frac{2}{\infty} = 0$

${\lim}_{x \to - \infty} \left(2 {e}^{-} x\right) = 2 {e}^{- - \infty} = 2 {e}^{\infty} = \infty$

This tells us that the graph asymptotes towards $y = 0$ as $x$ goes to positive infinite, which means that it approaches but never touches zero; and it tends to $y = \infty$ as $x$ goes towards negative infinite. Therefore, our range is positive real numbers, just as we thought.

There are no restrictions on the values $x$ can take, so the domain is all real numbers.