Problem 50 - Calculus: One and Several Variables

Question

Form the combinations \(f+g . f \quad-g . f\) \(g_{i} f / g\) and specify the domain of combination. $$f(x)=x^{2}-4, \quad g(x)=x+1 / x$$

Step-by-Step Solution

Answer

The domains of the combinations are: for \(f+g\) and \(f-g\), all real numbers except 0, and for \(f/g\), all real numbers except 0 and -1.

Step 1: Formulate the Combinations

Formulate the combinations as given, which are \(f+g\), \(f-g\), and \((f/g)\). Implement the given functions into these combinations. The functions given are \(f(x) = x^{2}-4\) and \(g(x) = x+1 / x\). \nSo, the combinations formed are:\n1. \(f+g = (x^{2}-4)+(x+1 / x) = x^{2}+x-4+1/x \)\n2. \(f-g = (x^{2}-4)-(x+1 / x) = x^{2}-x-4-1/x \)\n3. \(f/g = (x^{2}-4)/ (x+1 / x) = x(x^{2} -4)/(x+1)\).

Step 2: Determine the Domains

The domain of the combination of these functions is the set of x values for which the functions are defined.\n1. The domain of \(f+g\) and \(f-g\) are all real numbers except 0, as we cannot divide by zero (from the term 1/x in g).\n2. For the combination \(f/g\), the denominator cannot be zero. Therefore, the domain will be \(x ( x +1) != 0\). After solving, we get the values x != 0 and x != -1. So, the domain of \(f/g\) will be all real numbers excluding 0 and -1.

Key Concepts

Function Domain

Understanding the domain of a function is a fundamental concept in calculus, pivotal for working with any type of function. The domain refers to the set of all possible input values (usually represented by x) that a function can accept without causing any mathematical inconsistencies, such as division by zero or the square root of a negative number in the real number system.

For instance, in the exercise provided, the function g(x) = x + 1/x has a domain restriction because division by zero is undefined. Therefore, for g(x), all real numbers except x = 0 are permissible values. Similarly, when combining functions through operations like addition, subtraction, or division, we need to consider the domains of all participating functions and identify any additional restrictions. For the given function combinations, for example, the domain of f/g excludes x = -1 in addition to x = 0, because these values would make the denominator equal to zero in the simplified rational expression, leading to an undefined result. It is vital to ensure that the domain reflects the complete set of values for which the function combination is defined.

Function Operations

When working with function operations such as addition, subtraction, multiplication, or division, we are essentially combining two or more functions to create a new function. The key is to perform the operations term by term, following the standard arithmetic rules.

In the provided exercise, we combined the two functions f(x) and g(x) using different operations. For addition, we added the corresponding terms of f(x) = x^2 - 4 and g(x) = x + 1/x arriving at f + g = x^2 + x - 4 + 1/x. The subtraction followed a similar process, giving f - g = x^2 - x - 4 - 1/x. When dividing one function by another, we placed f(x) in the numerator and g(x) in the denominator and simplified the resulting expression. The focus of function operations is not only in carrying out the process correctly but also in understanding how these operations influence the domain of the resulting function.

Rational Functions

Rational functions are ratios of two polynomial functions. They take the form (P(x)/Q(x)), where both P(x) and Q(x) are polynomials and Q(x) is not the zero polynomial. The characteristics of rational functions are deeply influenced by their domains since the values which make the denominator zero must be excluded from the domain.

Rational expressions can be simplified, factored, or expanded to explore their properties further and to simplify calculations like determining their domain. For example, in the combination f/g from the exercise, the initial rational function (x^2 - 4)/(x + 1/x) was simplified by multiplying the numerator and denominator by x, to further define its domain and for ease of understanding. Identifying the excluded values in the domain, such as x = 0 and x = -1 for f/g, helps preemptively address potential points of discontinuity, which are significant features for rational functions.