# How do you find the domain and range of e^(-4t)?

Nov 21, 2017

Domain: All Reals
Range: Y>0

#### Explanation:

Start by finding the Domain (t values) and then use that information to solve for the Range (y values)

for ${e}^{- 4 t}$ look for the number t cannot be.

Look for things such as:
-Dividing by 0
-Taking a root of a negative number
-Taking the $\ln$ of a negative number

${e}^{- 4 t} = \frac{1}{{e}^{4 t}}$

Now the only issue is that the denominator cannot equal 0.
But $e$ (being a positive number) to any power will be a number greater than 0 and so there are no exceptions for $t$.

Therefore the Domain is all real numbers.

The Range would then be the results of all the values of $t$.

When t is really really big the equation gets closer and closer to 0.
$\left(\frac{1}{10000000} = 0.0000001\right)$ As $t$ gets bigger the the result gets closer and closer still to zero, but never equals zero. So its always positive as $t$ gets larger.

When $t$ is small (less than 0) you get ${e}^{-} \left(4 \left(- t\right)\right)$ or simply ${e}^{4 t}$
So as $t$ increases the result goes to infinity. So again we get all positive numbers.

Therefore the range is all values greater than 0.

You can also see this graphically.

graph{e^(-4x) [-10, 10, -10,10]}