Problem 74 - Calculus Early Transcendentals

Question

Find the domains of the following functions. Specify the domain mathematically and then describe it in words or with a sketch. $$g(x, y, z)=\frac{10}{x^{2}-(y+z) x+y z}.$$

Step-by-Step Solution

Answer

Answer: The domain of the function g(x, y, z) is the set of all points (x, y, z) in the three-dimensional coordinate system such that $$(-y-z)^2 - 4(yz) \neq 0.$$

Step 1: Examine the Denominator

First, we need to find when the denominator of g(x, y, z) equals zero. The denominator of the given function is: $$x^2 - (y+z)x + yz$$ Set the denominator equal to zero: $$x^2 - (y+z)x + yz = 0$$

Step 2: Solve for x

Now, we need to solve the equation for x. The equation is a quadratic equation in the form of: $$x^2 - (y+z)x + yz = 0$$ Apply the Quadratic Formula: $$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$$ In this case, a = 1, b = -(y+z), and c = yz. Plugging these values into the Quadratic Formula: $$x = \frac{y+z \pm \sqrt{(-y-z)^2-4(yz)}}{2}$$

Step 3: Examine the Discriminant

In order to determine when the function is undefined, we need to examine the discriminant, which is the part of the quadratic formula inside the square root: $$\Delta = (-y-z)^2 - 4(yz)$$ If the discriminant equals zero, it means that the function is undefined for some values of x, y, and z. We are looking for when the function will be defined, so we want the discriminant to be nonzero: $$(-y-z)^2 - 4(yz) \neq 0$$

Step 4: Determine the Domain

Since we want the discriminant to be nonzero, we find the domain of g(x, y, z) to be: $$\{ (x, y, z) \mid (-y-z)^2 - 4(yz) \neq 0 \} $$

Step 5: Describe the Domain

The domain of the function g(x, y, z) consists of all values of x, y, and z such that the discriminant is nonzero. This implies that for any values of x, y, and z, the function g(x, y, z) will be well-defined, as long as the discriminant remains nonzero. Since this condition involves multiple variables, it is difficult to provide an accurate graphical representation of the domain. However, we can verbally describe the domain as the set of all points (x, y, z) in the three-dimensional coordinate system satisfying the condition $$(-y-z)^2 - 4(yz) \neq 0.$$

Key Concepts

Quadratic Formula

The quadratic formula is a powerful mathematical tool used to find the solutions of quadratic equations. A quadratic equation is typically expressed in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The quadratic formula allows you to solve for \( x \) using the expression:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Here’s what each part of the formula represents:

• \( b^2 - 4ac \) is called the "discriminant." It determines the nature of the roots.
• The "\( \pm \)" symbol indicates that there are usually two solutions: one using the plus and one using the minus.

This formula is extremely helpful for finding the roots of the quadratic equation which come up in various areas of mathematics and science. Remember that both real and complex numbers can be solutions, depending on the value of the discriminant.
Understanding the quadratic formula is essential when solving problems involving parabolas, projectiles, and other phenomena described by quadratic equations.

Discriminant

The discriminant is a component of the quadratic formula and plays a crucial role in determining the nature of the roots of a quadratic equation. It is the term under the square root sign, \( \Delta = b^2 - 4ac \). Here’s how the discriminant affects the solutions of the quadratic equation:

• If \( \Delta > 0 \), the quadratic equation has two distinct real roots. This means the parabola crosses the x-axis at two different points.
• If \( \Delta = 0 \), the quadratic equation has exactly one real root or a repeated root. The parabola touches the x-axis at one point, known as the vertex.
• If \( \Delta < 0 \), the quadratic equation has two complex conjugate roots. Graphically, this means the parabola does not intersect the x-axis.

In multivariable functions such as \( g(x, y, z) \), the discriminant determines when the denominator could become zero, affecting the domain of the function. It’s important to ensure that the discriminant is nonzero to keep the function defined within its domain.
Thus, analyzing the discriminant gives insight into the behavior of quadratic equations and helps predict whether solutions will be real or complex.

Three-Dimensional Coordinate System

The three-dimensional coordinate system is an extension of the familiar two-dimensional system into a space that accounts for width, depth, and height. It is an essential tool in representing points in a 3D space using three coordinates, typically denoted as \( (x, y, z) \).

• The \( x \)-axis typically runs horizontally and represents width.
• The \( y \)-axis usually goes vertically, representing height or depth.
• The \( z \)-axis adds the third dimension, accommodating height when combined with the xy-plane.

This system allows us to describe locations, vectors, and functions in three-dimensional space. For functions like \( g(x, y, z) \), the domain is all the points in this 3D space where the function remains defined.
When dealing with such functions, it's crucial to remember that they can exhibit more complex behaviors compared to their two-dimensional counterparts. Visualization and analysis in three-dimensional space are important for fields such as physics, engineering, and computer graphics, where spatial relationships and structures are more intricate.