Problem 2 - Algebra and Trigonometry

Question

\(1-6\) Find \(f+g, f-g, f g,\) and \(f / g\) and their domains. $$ f(x)=x^{2}+2 x, \quad g(x)=3 x^{2}-1 $$

Step-by-Step Solution

Answer

Sum: \(4x^2 + 2x - 1\), Difference: \(-2x^2 + 2x + 1\), Product: \(3x^4 + 6x^3 - x^2 - 2x\), Quotient: \(\frac{x^2 + 2x}{3x^2 - 1}\); Domains: All real numbers except for the quotient, which excludes \(\pm\frac{1}{\sqrt{3}}\).

Step 1: Finding the Sum

To find \((f+g)(x)\), add the functions \(f(x)\) and \(g(x)\). Thus, \((f+g)(x) = (x^2 + 2x) + (3x^2 - 1)\). Combine like terms: \((f+g)(x) = 4x^2 + 2x - 1\).

Step 2: Finding the Domain of the Sum

For the domain of \((f+g)(x) = 4x^2 + 2x - 1\), since there are no restrictions for the variables in a polynomial, the domain is all real numbers \((-fty, fty)\).

Step 3: Finding the Difference

To find \((f-g)(x)\), subtract \(g(x)\) from \(f(x)\). Thus, \((f-g)(x) = (x^2 + 2x) - (3x^2 - 1)\). Combine like terms: \((f-g)(x) = -2x^2 + 2x + 1\).

Step 4: Finding the Domain of the Difference

The domain of \((f-g)(x) = -2x^2 + 2x + 1\), is also all real numbers \((-fty, fty)\) since it is a polynomial.

Step 5: Finding the Product

To find \((f \, g)(x)\), multiply the functions \(f(x)\) and \(g(x)\). Thus, \((f \, g)(x) = (x^2 + 2x)(3x^2 - 1)\). Expand this to get \(3x^4 - x^2 + 6x^3 - 2x\).

Step 6: Finding the Domain of the Product

The domain of \((f \, g)(x) = 3x^4 + 6x^3 - x^2 - 2x\) is all real numbers \((-fty, fty)\) because it is a polynomial.

Step 7: Finding the Quotient

To find \((f/g)(x)\), divide \(f(x)\) by \(g(x)\). Thus, \( (f/g)(x) = \frac{x^2 + 2x}{3x^2 - 1} \).

Step 8: Finding the Domain of the Quotient

The domain of \( (f/g)(x) \) is all real numbers except where the denominator is zero. Solve \(3x^2 - 1 = 0\) to find the restriction. \(x^2 = 1/3\), so \(x = \pm\frac{1}{\sqrt{3}}\). The domain is \((-fty, -\frac{1}{\sqrt{3}}) \cup (-\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}) \cup (\frac{1}{\sqrt{3}}, fty)\).

Key Concepts

Domain of a Function

The domain of a function is critical because it tells us all the possible input values (usually denoted as "x") that a function can have without encountering undefined expressions. Think of it as the set of all possible x-values that will allow the function to work without any issues.
For polynomials like \[f(x) = 4x^2 + 2x - 1\], there are no restrictions on x because polynomials are defined for all real numbers. Thus, their domain is \((-\infty, \infty)\).

• No division by zero is involved.
• No even roots of negative numbers are present.
• All real numbers are included.

In contrast, for rational functions such as \((f/g)(x) = \frac{x^2 + 2x}{3x^2 - 1}\), the denominator must not be zero. Solving \(3x^2 - 1 = 0\) shows x-values where the function is undefined.
This creates a restricted domain. In this case, x cannot be \(\pm\frac{1}{\sqrt{3}}\), so the domain excludes these x-values:
\[(-\infty, -\frac{1}{\sqrt{3}}) \cup (-\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}) \cup (\frac{1}{\sqrt{3}}, \infty)\].

Polynomial

Polynomials are powerful tools in mathematics. They are expressions involving variables raised to whole number exponents, combined using addition, subtraction, and multiplication.

For example, \[f(x) = 4x^2 + 2x - 1\] is a polynomial function where each term involves a varying power of x.

• The highest power of x indicates the degree of the polynomial.
• In \[f(x)\], the degree is 2.
• This often affects the shape and number of turning points on the graph.

Polynomials can be simple, like a linear expression (degree 1) or as complex as desired with higher degrees.
The beauty of polynomials includes:

• They are continuous.
• They are smooth, without holes or breaks.
• They extend infinitely in both directions.

These traits make polynomials an essential subject in understanding the wide behavior of many physical and theoretical phenomena.

Rational Functions

Rational functions are another class of functions that involve the ratio of two polynomials. They have the form \(\frac{P(x)}{Q(x)}\), where both P(x) and Q(x) are polynomial functions.
Our example, \(\frac{x^2 + 2x}{3x^2 - 1}\), is a rational function.
The key feature of rational functions often revolves around their undefined points, usually at the values of x making the denominator zero. These points are values where the function "breaks" and often lead to vertical asymptotes in the graph.

• Rational functions can have horizontal or slant asymptotes.
• They can exhibit different behaviors and limitations compared to other function types.

Understanding rational functions means recognizing these character traits:

• The domain is usually all real numbers except those causing division by zero.
• They are versatile in representing real-world problems where ratios naturally occur.

Rational functions' intriguing mix of continuous and discontinuous behavior makes them a fascinating area of study in algebra.