Problem 6 - College Algebra with Corequisite Support

Question

For the following exercises, determine the domain for each function in interval notation. Given \(f(x)=-3 x^{2}+x\) and \(g(x)=5,\) find \(f+g, \quad f-g, \quad f g,\) and \(\quad \frac{f}{g}\)

Step-by-Step Solution

Answer

All combined functions (f+g, f-g, fg, and \(\frac{f}{g}\)) have the domain \((-\infty, \infty)\).

Step 1: Understand the Problem

You need to find the domain for the functions obtained by combining the functions \(f(x) = -3x^2 + x\) and \(g(x) = 5\) through addition, subtraction, multiplication, and division. Then express these domains in interval notation.

Step 2: Determine Domain of f(x) and g(x)

Since \( f(x) = -3x^2 + x \) is a polynomial function, its domain is all real numbers, expressed as \((-\infty, \infty)\). Similarly, \( g(x) = 5 \) is a constant function and has the same domain of \((-\infty, \infty)\).

Step 3: Find f+g and its Domain

To find \((f+g)(x)\), add \(f(x)\) and \(g(x)\): \((-3x^2 + x) + 5 = -3x^2 + x + 5\). The domain remains \((-\infty, \infty)\) as it is also a polynomial function.

Step 4: Find f-g and its Domain

Subtract \(g(x)\) from \(f(x)\): \((-3x^2 + x) - 5 = -3x^2 + x - 5\). This function is a polynomial, so its domain is \((-\infty, \infty)\).

Step 5: Find fg and its Domain

Multiply \(f(x)\) and \(g(x)\): \((-3x^2 + x)(5) = -15x^2 + 5x \). This function is also a polynomial, so its domain is \((-\infty, \infty)\).

Step 6: Find \(\frac{f}{g}\) and its Domain

Divide \(f(x)\) by \(g(x)\): \(\frac{-3x^2 + x}{5}\). This results in a polynomial divided by a constant, so the domain is \((-\infty, \infty)\) since \(g(x)\) does not equal zero, allowing division.

Key Concepts

Polynomial Functions

Polynomial functions are one of the most fundamental types of functions in mathematics. A polynomial function is constructed from variables and constants using only operations of addition, subtraction, multiplication, and non-negative integer exponents. Generally, a polynomial function can be expressed in the form:

• \( P(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \)

where the \( a_i \) are constants and \( n \) is a non-negative integer known as the degree of the polynomial.
Notably, the domain of a polynomial function is always all real numbers, \((-\infty, \infty)\), because there are no restrictions on \(x\) for any real number input, unlike functions involving division by zero or taking square roots of negative numbers. This makes them easy to work with in terms of domain analysis.
Polynomials are smooth and continuous, which means there are no breaks, holes, or jumps in their graphs. Whether you're working with a simple linear function \(f(x) = x\) or a more complex quadratic like \(f(x) = -3x^2 + x\), understanding polynomials helps in tackling a variety of mathematical problems.

Interval Notation

Interval notation is a concise way to represent subsets of real numbers, which makes it particularly useful for describing domains of functions.
Here is a step-by-step guide on how to read and write interval notation:

• Use parentheses \(()\) for intervals that do not include their endpoints and brackets \([]\) for intervals that do.
• An interval like \((-\infty, \infty)\) indicates that every real number is included, but \(-\infty\) and \(\infty\) themselves are not real numbers, so they are always paired with parentheses.
• If an interval looks like \([a, b)\), it means that \(a\) is included in the interval, but \(b\) is not.

When finding the domain of functions, the use of interval notation is especially helpful. For instance, many polynomial functions like \(f(x) = -3x^2 + x\) have domains that extend over all real numbers, which is neatly captured in the interval notation \((-\infty, \infty)\).
Understanding this notation is crucial for accurately communicating possible values that can be input into a function.

Operations on Functions

Operations on functions include addition, subtraction, multiplication, and division of two or more functions. Let's dive into each operation:

• Addition (\(f+g\)): Combine \(f(x)\) and \(g(x)\) by adding: \((f+g)(x) = f(x) + g(x)\)
• Subtraction (\(f-g\)): Find the difference with \((f-g)(x) = f(x) - g(x)\)
• Multiplication (\(fg\)): Multiply the functions: \((fg)(x) = f(x) \cdot g(x)\)
• Division (\(\frac{f}{g}\)): Divide \(f(x)\) by \(g(x)\), with the caveat that \(g(x) eq 0\): \(\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}\)

In the given exercise, polynomial functions \(-3x^2 + x\) and the constant function \(5\) form the basis for these operations. For example, \(f+g\) results in another polynomial \(-3x^2 + x + 5\) with a domain of \((-\infty, \infty)\). Similarly, \(f-g\) and \(fg\) also retain polynomial characteristics.
Division, \(\frac{f}{g}\), is straightforward in this context, as dividing by a non-zero constant \(g(x) = 5\) means the domain remains \((-\infty, \infty)\). These operations show that combining functions doesn’t usually change the domain if all components have the same domain.