Data 605 Discussion 15

Exercise 23

For the function \(f(x, y, z) = \frac{x}{x + 2y - 4z}\), the domain and range are as follows:

Domain:
The function is defined for all real numbers \(x\), \(y\), and \(z\) such that the denominator \(x + 2y - 4z\) is not equal to zero. Thus, the domain can be expressed as:

\[ \text{Domain} = \{(x, y, z) \in \mathbb{R}^3 \mid x + 2y - 4z \neq 0\} \]

Range:
To find the range, we need to examine the behavior of the function as the denominator approaches zero. However, the range is not immediately evident without further analysis, as it depends on the specific values of \(x\), \(y\), and \(z\). Therefore, the range is not explicitly stated without additional context.

Exercise 24

For the function \(f(x, y, z) = \frac{1}{1 - x^2 - y^2 - z^2}\), the domain and range are as follows:

Domain:
Similar to the previous function, the function is defined for all real numbers \(x\), \(y\), and \(z\) such that the denominator \(1 - x^2 - y^2 - z^2\) is not equal to zero. Hence, the domain can be expressed as:

\[ \text{Domain} = \{(x, y, z) \in \mathbb{R}^3 \mid 1 - x^2 - y^2 - z^2 \neq 0\} \]

Range:
As the denominator approaches zero, the function tends towards infinity or negative infinity. Therefore, the range is all real numbers except zero, which can be expressed as:

\[ \text{Range} = \mathbb{R} \setminus \{0\} \]

Exercise 25

For the function \(f(x, y, z) = \sqrt{z - x^2 + y^2}\), the domain and range are as follows:

Domain:
The function is defined for all real numbers \(x\), \(y\), and \(z\) such that the expression inside the square root \(z - x^2 + y^2\) is non-negative. Thus, the domain can be expressed as:

\[ \text{Domain} = \{(x, y, z) \in \mathbb{R}^3 \mid z - x^2 + y^2 \geq 0\} \]

Range:
Since the square root function outputs non-negative values, the range of the function will be all non-negative real numbers, including zero. Therefore, the range can be expressed as:

\[ \text{Range} = [0, \infty) \]

Exercise 26

For the function \(f(x, y, z) = z^2 \sin{x} \cos{y}\), the domain and range are as follows:

Domain:
The function is defined for all real numbers \(x\), \(y\), and \(z\) without any restrictions on their values. Hence, the domain is the set of all real numbers:

\[ \text{Domain} = \mathbb{R}^3 \]

Range:
The range of the function is determined by the behavior of \(\sin{x}\) and \(\cos{y}\), which oscillate between -1 and 1 for all real inputs. Since \(z^2\) can take any non-negative value, the range of the function will be all real numbers between \(-z^2\) and \(z^2\). Therefore, the range can be expressed as:

\[ \text{Range} = [-z^2, z^2] \]