# How do you find the domain and range of f(x)=2-2^x?

Jul 19, 2016

$- \infty < x < + \infty$
$- \infty < f \left(x\right) < 2$

#### Explanation:

Function $y = {2}^{x}$ is defined for all real $x$.
It is always positive.
It is monotonically increasing .
If $x$ goes to $- \infty$, it asymptotically decreasing to $0$.
If $x$ goes to $+ \infty$, it increasing to $+ \infty$.

Therefore, function $y = - {2}^{x}$ has the following properties:
It is defined for all real $x$.
It is always negative.
It is monotonically decreasing .
If $x$ goes to $- \infty$, it asymptotically increasing to $0$.
If $x$ goes to $+ \infty$, it decreasing to $- \infty$.

Next step is to add $2$ to this function to get $y = 2 - {2}^{x}$.
It has the following properties:
It is defined for all real $x$.
It is always less than $2$
It is monotonically decreasing .
If $x$ goes to $- \infty$, it asymptotically increasing to $2$.
If $x$ goes to $+ \infty$, it decreasing to $- \infty$.

We might suggest to use educational materials on UNIZOR.COM to study functions and their behavior. Exponential functions are explained there in lots of detail.