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

Jul 8, 2015

f(x) : RR -> ]-oo;2[

#### Explanation:

$f \left(x\right) = 2 - {e}^{\frac{x}{2}}$

Domain : ${e}^{x}$ is defined on $\mathbb{R}$.
And ${e}^{\frac{x}{2}} = {e}^{x \cdot \frac{1}{2}} = {\left({e}^{x}\right)}^{\frac{1}{2}} = \sqrt{{e}^{x}}$ then ${e}^{\frac{x}{2}}$ is defined on $\mathbb{R}$ too.

And so, the domain of $f \left(x\right)$ is $\mathbb{R}$

Range :

The range of ${e}^{x}$ is ${\mathbb{R}}^{+} - \left\{0\right\}$.

Then :

$0 < {e}^{x} < + \infty$
$\iff \sqrt{0} < \sqrt{{e}^{x}} < + \infty$
$\iff 0 < {e}^{\frac{x}{2}} < + \infty$
$\iff 0 > - {e}^{\frac{x}{2}} > - \infty$
$\iff 2 > 2 - {e}^{\frac{x}{2}} > - \infty$

Therefore,
$\iff 2 > f \left(x\right) > - \infty$