Question

What is the domain and range of `f(x) = 2 - e ^ (x / 2)`?

Answer 1

Domain: `(-oo,oo)`

Range: `(-oo,2)`

Explanation:

The domain is all possible values of `x` with which `f(x)` is defined.

Here, any value of `x` will result in a defined function. Therefore, the domain is `-oo<` `x<` `oo`, or, in interval notation:

`(-oo,oo)`.

The range is all possible values of `f(x)`. It can also be defined as the domain of `f^-1(x)`.

So to find `f^-1(x):`

`y=2-e^(x/2)`

Interchange the variables `x` and `y`:

`x=2-e^(y/2)`

And solve for `y`:

`x-2=-e^(y/2)`

`e^(y/2)=2-x`

Take the natural logarithm of both sides:

`ln(e^(y/2))=ln(2-x)`

`y/2ln(e)=ln(2-x)`

As `ln(e)=1`,

`y/2=ln(2-x)`

`y=2ln(2-x)=f^-1(x)`

We must find the domain of the above.

For any `lnx,` `x>0`.

So here, `2-x>0`

`-x> -2`

`x` `<` `2`

So the range of `f(x)` can be stated as `(-oo,2)`