Q 51. - Calculus

Question

Determine the domains of the functions in Exercises 47–56, and find where the functions are continuous.  

$f(x,y,z)=\frac{\sin ( x+y+z)}{\sqrt{x+y+z}}$

Step-by-Step Solution

Answer

The function $f(x,y,z)=\frac{\sin ( x+y+z)}{\sqrt{x+y+z}}$ is continuous over its entire domain. 

Step 1. Given information.

We have given expression: $f(x,y,z)=\frac{\sin ( x+y+z)}{\sqrt{x+y+z}}$

Step 2: To find continuous of the function.

The given function is a rational expression of sine function and a radical function. 

The domain of a sine function is the entire set of real numbers. 

The radical term in the denominator makes sure that the term inside the radical sign is greater than or equal to 0. 

That is  $x+y+z\ge 0$

The radical expression means the denominator is not equal to $0$.

$$\begin{gathered} \sqrt{x+y+z}\ne 0 \\ x+y+z\ne 0 \end{gathered}$$

The domain of the given function:

$Domain\left(f\right)=\left\{\left(x,y,z\right):x+y+z>0\right\}$

A rational expression is said to be continuous over its domain always.