Question

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

Answer 1

`-oo < x < +oo`
`-oo < f(x) < 2`

Explanation:

Function `y = 2^x` is defined for all real `x`.
It is always positive.
It is monotonically increasing .
If `x` goes to `-oo`, it asymptotically decreasing to `0`.
If `x` goes to `+oo`, it increasing to `+oo`.

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 `-oo`, it asymptotically increasing to `0`.
If `x` goes to `+oo`, it decreasing to `-oo`.

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 `-oo`, it asymptotically increasing to `2`.
If `x` goes to `+oo`, it decreasing to `-oo`.

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.