Problem 7 - College Algebra with Corequisite Support

Question

For the following exercises, determine the domain for each function in interval notation. Given \(f(x)=2 x^{2}+4 x\) and \(g(x)=\frac{1}{2 x},\) find \(f+g, \quad f-g, \quad f g,\) and \(\frac{f}{g}\)

Step-by-Step Solution

Answer

The domain for all combined functions \(f+g\), \(f-g\), \(f \cdot g\), and \(\frac{f}{g}\) is \((-\infty, 0) \cup (0, +\infty)\).

Step 1: Determine the Domain of f(x)

The function \(f(x) = 2x^2 + 4x\) is a polynomial. Polynomial functions are defined for all real numbers, so the domain of \(f(x)\) is \((-\infty, +\infty)\).

Step 2: Determine the Domain of g(x)

The function \(g(x) = \frac{1}{2x}\) has a restriction where the denominator cannot be zero. Thus, \(2x eq 0\) leads to \(x eq 0\). The domain of \(g(x)\) is \((-\infty, 0) \cup (0, +\infty)\).

Step 3: Find the Domain of f+g

The domain of \(f+g\) is the intersection of the domains of \(f(x)\) and \(g(x)\), which is \((-\infty, 0) \cup (0, +\infty)\).

Step 4: Find the Domain of f-g

The domain of \(f-g\) is also the intersection of the domains of \(f(x)\) and \(g(x)\), which is \((-\infty, 0) \cup (0, +\infty)\).

Step 5: Find the Domain of f*g

The domain of \(f \cdot g\) is the intersection of the domains of \(f(x)\) and \(g(x)\), which is \((-\infty, 0) \cup (0, +\infty)\).

Step 6: Find the Domain of f/g

The domain of \(\frac{f}{g}\) involves the intersection of the domains of \(f(x)\) and \(g(x)\) while ensuring \(g(x) eq 0\). Since \(g(x) = \frac{1}{2x}\), \(x eq 0\) already, so the domain is \((-\infty, 0) \cup (0, +\infty)\).

Step 7: Conclusion: Domains of Combined Functions

For \(f+g\), \(f-g\), \(fg\), and \(\frac{f}{g}\), the domains all are \((-\infty, 0) \cup (0, +\infty)\).

Key Concepts

Polynomial Functions

Polynomial functions are a fundamental concept in mathematics. They consist of terms that are made up of variables raised to whole number exponents, combined using addition, subtraction, and multiplication. A simple form of a polynomial function is given as \(f(x) = ax^n + bx^{n-1} + \, ... \, + z\), where \(a, b, ..., z\) are constants and \(n\) is a non-negative integer.

These functions are known for their smooth and continuous graphs. They can have one or more terms and do not involve division by a variable. This characteristic gives polynomial functions an unrestricted domain, meaning they are defined for every real number.

Some key characteristics of polynomial functions include:

• They have no breaks, holes, or asymptotes.
• Their graphs are a series of smooth, continuous curves.
• The domain of a polynomial function is always \((-\infty, +\infty)\).

Rational Functions

Rational functions are the ratio of two polynomial functions. They take the form \(g(x) = \frac{p(x)}{q(x)}\), where both \(p(x)\) and \(q(x)\) are polynomials. The domain of a rational function is generally all real numbers except where the denominator, \(q(x)\), is equal to zero.

Understanding rational functions is crucial due to their asymptotic behavior—where the graph approaches a line but never actually touches it—creating vertical asymptotes where the denominator is zero.

Points of interest when analyzing rational functions include:

• The domain excludes any \(x\) values that make the denominator zero.
• There may be vertical asymptotes at the points where \(q(x)\) equals zero.
• Understanding the behavior near these asymptotes is important to analyze the entire graph.

Interval Notation

Interval notation is a method used to describe the domain of a function in a concise and standardized way. It uses brackets and parentheses to depict intervals of values on the number line.

Here’s how it works:

• Parentheses \(( )\) denote that an endpoint is not included in the interval, often used for infinity or points where the function is undefined.
• Brackets \([ ]\) mean the endpoint is included in the interval.
• For example, the domain \((-\infty, 0) \cup (0, +\infty)\) indicates all numbers except zero, where each part of the domain is separated by a union \(\cup\).

Interval notation is an efficient way to express complex domains, especially when dealing with non-continuous functions like the rational function described in the original problem.