Problem 39 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the domain of each function. (a) \(f(w, x, y, z)=\frac{1}{\sqrt{w^{2}+x^{2}+y^{2}+z^{2}}}\) (b) \(g\left(x_{1}, x_{2}, \ldots, x_{n}\right)=\exp \left(-x_{1}^{2}-x_{2}^{2}-\cdots-x_{n}^{2}\right)\) (c) \(h\left(x_{1}, x_{2}, \ldots, x_{n}\right)=\sqrt{1-\left(x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}\right)}\)

Step-by-Step Solution

Answer

(a) All real numbers except (0, 0, 0, 0). (b) All real numbers. (c) Vectors such that the sum of squares \(\leq 1\).

Step 1: Analyze Function (a)

The function is given by \[ f(w, x, y, z)=\frac{1}{\sqrt{w^{2}+x^{2}+y^{2}+z^{2}}} \] The expression under the square root must be greater than zero, i.e., \[ w^2 + x^2 + y^2 + z^2 > 0 \] This means the vector \((w, x, y, z)\) cannot be the zero vector \((0, 0, 0, 0)\). Thus, the domain is all real numbers, excluding the point where all variables are zero.

Step 2: Determine Domain of Function (b)

The function \[ g\left(x_{1}, x_{2}, \ldots, x_{n}\right)=\exp\left(-x_{1}^{2}-x_{2}^{2}-\cdots-x_{n}^{2}\right) \] includes an exponential function where the exponent is quadratic. Since exponential functions are defined for all real numbers, no restrictions are imposed by the quadratic terms. Therefore, the domain is all real numbers for each variable \(x_1, x_2, \ldots, x_n\).

Step 3: Examine Domain of Function (c)

The function \[ h\left(x_{1}, x_{2}, \ldots, x_{n}\right)=\sqrt{1-\left(x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}\right)} \] requires that the expression under the square root is non-negative, i.e., \[ 1 - (x_{1}^{2} + x_{2}^{2} + \cdots + x_{n}^{2}) \geq 0 \] Simplifying gives \[ x_{1}^{2} + x_{2}^{2} + \cdots + x_{n}^{2} \leq 1 \] Thus, the domain is the set of all vectors \(x_1, x_2, \ldots, x_n\) such that their squared sum does not exceed 1.

Key Concepts

Multivariable Functions

In mathematics, a multivariable function is like a magical formula that takes several variables as input and produces a single output. These variables could be anything, like time, position, or temperature, depending on the situation you're dealing with. Imagine using different ingredients to cook a recipe; each ingredient plays a role in the final dish, just like each variable influences the outcome of a multivariable function.

To understand this type of function, let's consider function \(f(w, x, y, z)=\frac{1}{\sqrt{w^{2}+x^{2}+y^{2}+z^{2}}}\). This function relies on four different variables, each contributing to the outcome. The input here is a combination of these variables, analogous to combining different colors to create a painting.

The key challenge with multivariable functions is determining their domain, which is the set of all possible inputs that result in a valid output. For our first function, the domain excludes points where the contribution from all variables is zero. Imagine trying to divide by zero, which is mathematically impossible, so we avoid such scenarios.

Square Roots in Functions

Square roots in functions often require special consideration, as they impose specific conditions on the domain of the function. The square root expression \(\sqrt{something}\) must always be non-negative, meaning that ‘something’ must be zero or positive. If not, the square root isn’t valid under real numbers.

Looking at the function \(h\left(x_{1}, x_{2}, \ldots, x_{n}\right)=\sqrt{1-\left(x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}\right)}\), we see that the term inside the square root must satisfy \(1 - (x_{1}^{2} + x_{2}^{2} + \cdots + x_{n}^{2}) \geq 0\), which implies that the sum of squares of the variables must be less than or equal to 1.

• This setup often describes a geometric shape, such as a circle or sphere in higher dimensions.
• If we envision this in two dimensions, the domain can be seen as the inside of a circle on a coordinate plane, where any point (\(x_1, x_2\)) lies on or within the circle of radius 1.

This visualization helps us understand why certain inputs are included or excluded in the domain, especially in functions that involve square roots.

Exponential Functions Domain

Exponential functions are remarkable for their unique properties and are used to describe growth and decay processes in both natural and financial contexts. These functions are defined as \(\exp(y) = e^y\), where \(e\) is an important mathematical constant approximately equal to 2.718.

They boast a universal domain: these functions accept any real number as input. This means you can consider all real numbers for function \(g\left(x_{1}, x_{2}, \ldots, x_{n}\right)=\exp\left(-x_{1}^{2}-x_{2}^{2}-\cdots-x_{n}^{2}\right)\). In this case, even though the exponent in the exponential function includes a sum of squares, these terms do not restrict the domain.

• The negative sign means the exponential function decreases as the sum of squares increases, but this does not stop the evaluation of the function for any real numbers.
• It's like having a machine that works for any input thread, producing seamless fabrics of data, regardless of how complex the input pattern is.

Understanding this boundless domain can be incredibly helpful when modeling scenarios over continuous ranges and in scientific analyses where patterns need to be examined across vast datasets.