Consider the function $z=e^x\cos \left(y\right)$. Which of the following best describes the domain of this function?

• A. All real numbers $x$ such that $x>0$, and all real numbers $y$
• B. All real numbers $x$ and all real numbers $y$
• C. All real numbers $x$ such that $x\ge 0$, and all real numbers $y$
• D. All real numbers $x$ and $y$ such that $y\ne 0$

Answer: (B)

Verified step by step guidance

1. Step 1: Understand the function z = e^x * cos(y). This function is composed of two parts: the exponential function e^x and the cosine function cos(y). Both of these functions are well-defined for all real numbers.
2. Step 2: Analyze the exponential function e^x. The exponential function is defined for all real values of x, meaning there are no restrictions on x.
3. Step 3: Analyze the cosine function cos(y). The cosine function is also defined for all real values of y, meaning there are no restrictions on y.
4. Step 4: Combine the results from Step 2 and Step 3. Since both e^x and cos(y) are defined for all real numbers, the product z = e^x * cos(y) is also defined for all real numbers x and y.
5. Step 5: Conclude that the domain of the function z = e^x * cos(y) is all real numbers x and all real numbers y. This means there are no restrictions on the values of x or y.