# What is the domain and range of f(x,y) = 3 + sin(sqrt y-e^x)?

Feb 6, 2018

Range: $\left\{f \left(x , y\right) \in \mathbb{R} : 2 \le f \left(x , y\right) \le 4\right\}$
Domain: $\left\{\left(x , y\right) \in {\mathbb{R}}^{2} : y \ge 0\right\}$

#### Explanation:

Assuming a real valued function, the range of the sine function is $- 1 \le \sin \left(u\right) \le 1$, therefore, $f \left(x , y\right)$ can vary from $3 \pm 1$ and the range is:

$\left\{f \left(x , y\right) \in \mathbb{R} : 2 \le f \left(x , y\right) \le 4\right\}$

The domain for y is restricted by the fact that the argument for the radical must be greater than or equal to zero:

$\left\{y \in \mathbb{R} : y \ge 0\right\}$

The value of x can be any real number:

$\left\{\left(x , y\right) \in {\mathbb{R}}^{2} : y \ge 0\right\}$