Problem 41 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Find \((f+g)(x),(f g)(x)\), and \(\left(\frac{f}{g}\right)(x)\), and find their domains. $$ f(x)=2 x+3 \text { and } g(x)=x^{2}-1 $$

Step-by-Step Solution

Answer

(f+g)(x) is \(x^{2} + 2x + 2\) with domain being all real numbers. (fg)(x) is \(2x^{3} +3x^{2} - 2x - 3\) also with domain being all real numbers. \(\left(\frac{f}{g}\right)(x) = \frac{2x+3}{x^{2}-1}\) with domain all real numbers except for 1 and -1.

Step 1: Finding (f+g)(x)

This is obtained by adding the two functions together - \(f(x) + g(x)\). Calculation is as follows: \((f+g)(x) = (2x+3) + (x^{2}-1) = x^{2} + 2x + 2\)

Step 2: Finding (fg)(x)

This is obtained by multiplying the two functions together - \(f(x) \cdot g(x)\). Calculation is as follows: \((f g)(x) = (2x+3) \cdot (x^{2}-1) = 2x^{3} +3x^{2} - 2x - 3\)

Step 3: Finding \(\left(\frac{f}{g}\right)(x)\)

This is obtained by dividing the first function by the second - \(\frac{f(x)}{g(x)}\). Calculation is as follows: \(\left(\frac{f}{g}\right)(x) = \frac{2x+3}{x^{2}-1}\)

Step 4: Determining the Domains

The domain of (f+g)(x) and (fg)(x) are all real numbers (since x can be any real number). The domain of \(\frac{f}{g}(x)\) is all real number but not equal to 1 or -1 (i.e. \(x^{2}-1\neq 0\) so that we are not dividing by zero)

Key Concepts

Polynomial Functions

Polynomial functions are expressions that consist of variables and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
For instance, in the exercise, we have two functions:

• \(f(x) = 2x + 3\) — a first-degree polynomial also known as a linear function.
• \(g(x) = x^2 - 1\) — a second-degree polynomial, commonly referred to as a quadratic function.

The operations of addition, multiplication, and division among these polynomials create new polynomial expressions. The degree of the resulting polynomial depends on the operations performed. For example, adding two polynomials of degree \(n\) results in a polynomial of degree \(n\), whereas multiplying them may increase the degree.

Domain of a Function

The domain of a function refers to the set of all possible input values \((x)\) for which the function is defined.
For polynomial functions like \((f+g)(x)\) and \((fg)(x)\), the domain is typically all real numbers, since polynomials are defined for every real \(x\).

• For \((f+g)(x) = x^2 + 2x + 2\), any real number can be used without restrictions.
• For \((fg)(x) = 2x^3 + 3x^2 - 2x - 3\), similarly, there are no restrictions on \(x\).

However, when you divide two functions as in \(\left(\frac{f}{g}\right)(x) = \frac{2x+3}{x^2-1}\), it's crucial to exclude any \(x\)-values that make the denominator zero. Here, \(x^2-1\) equals zero when \(x = 1\) or \(x = -1\), so these are excluded from the domain.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to make them easier to work with or to find a specific result.
In this exercise:

• **Addition:** \((f+g)(x) = (2x + 3) + (x^2 - 1) = x^2 + 2x + 2\). We simply combine like terms, adding coefficients of the same degree.
• **Multiplication:** \((fg)(x) = (2x + 3)(x^2 - 1)\). Here, the distributive property \((a+b)(c+d) = ac + ad + bc + bd\) is used to expand and simplify the expression.
• **Division:** \(\left(\frac{f}{g}\right)(x) = \frac{2x+3}{x^2-1}\). Unlike addition and multiplication, division requires paying attention to the restrictions to ensure the expression is defined for specific \(x\).

Proper algebraic manipulation is crucial for correctly solving expressions and identifying domains.