Question

How do I Find the domain and Range of the Function: Y = (2e^x) + 4

Answer 1

Lest start with the domain (which x values the function goes to):
Ask your self, does any value make the function invalid (check each part)?
Since 4 is a constant and 2e^x is defined everywhere, both parts check out, so the domain is all real numbers, or,
\[Domain={x \in \mathbb{R}}\]

Now let's go for the hard part.
The range is the all of the y-values the function maps to.
(lets ignore the 4 right now)
Take 2e^x, which has a minimum value that's just above zero, and a maximum at infinity. or, \[0<y <\infty\]
Now we just have the 4 to deal with, which simply displaces the answer up 4.
So we have\[4<y<\infty\]

Written in set notation:
\[Range=\left\{ y \in \mathbb{R}: 4<y<\infty \right\}\]