Problem 31 - College Algebra Essentials

Question

In Exercises \(31-50,\) find \(f+g, f-g,\) fg, and \(\frac{f}{g} .\) Determine the domain for each function. $$f(x)=2 x+3, g(x)=x-1$$

Step-by-Step Solution

Answer

The sum \(f+g\) is \(3x + 2\) with domain (-\infty, \infty), the difference \(f-g\) is \(x + 4\) with domain (-\infty, \infty), the product \(fg\) is \(2x^2 + x - 3\) with domain (-\infty, \infty), and the fraction \(f/g\) is \((2x + 3) / (x - 1)\) with domain (-\infty, 1) \cup (1, \infty).

Step 1: Add the functions \(f(x)\) and \(g(x)\)

To find the sum \(f+g\), add \(f(x)\) and \(g(x)\) together: \(f(x) + g(x) = (2x + 3) + (x - 1) = 3x + 2\).

Step 2: Subtract \(g(x)\) from \(f(x)\)

To find the difference \(f-g\), subtract \(g(x)\) from \(f(x)\): \(f(x) - g(x) = (2x + 3) - (x - 1) = x + 4\).

Step 3: Multiply the functions \(f(x)\) and \(g(x)\)

To find the product \(f \cdot g\), multiply \(f(x)\) and \(g(x)\): \(f(x) \cdot g(x) = (2x + 3) \cdot (x - 1) = 2x^2 + x - 3\).

Step 4: Divide \(f(x)\) by \(g(x)\)

To find the fraction \(f/g\), divide \(f(x)\) by \(g(x)\). Make sure that g(x) is not equal to 0 because division by zero is undefined: \(f(x) / g(x) = (2x + 3) / (x - 1)\). This function is defined for \(x \neq 1\).

Step 5: Determine the domain for each function

For our functions the domains are: \(f+g: (-\infty, \infty), f-g: (-\infty, \infty), fg: (-\infty, \infty), f/g: (-\infty, 1) \cup (1, \infty)\). The domain of the function \(f/g\) doesn't include 1 because at \(x = 1\) the function \(g(x) = 0\) which leads to division by zero.

Key Concepts

Domain of a Function

The domain of a function is a crucial concept in mathematics, particularly when working with different types of functions. It refers to the set of all possible input values (usually x-values) that a function can accept without resulting in undefined situations. Understanding the domain helps in knowing where a function is "valid" or can be realistically applied.

For instance, both the functions given in the exercise, \(f(x) = 2x + 3\) and \(g(x) = x - 1\), are polynomial functions. By nature, polynomial functions have a domain of all real numbers, which is written as \((−\infty, \infty)\). Therefore, the domain for the operations \(f+g, f-g,\) and \(fg\) also covers all real numbers.

However, when dealing with division, as in \(f/g\), the domain changes. This is because division by zero is undefined. In \((f/g)(x) = \frac{2x + 3}{x - 1}\), the function \(g(x) = x - 1\) becomes zero when \(x = 1\). Therefore, \(x\) cannot be 1. In this case, the domain is \((−\infty, 1) \cup (1, \infty)\), which excludes this point but includes all other real numbers.

Polynomial Functions

Polynomial functions are one of the most fundamental types of functions in algebra. They are composed of variables and coefficients, involving only operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. The general form of a polynomial function in one variable \(x\) is \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where \(a_n, a_{n-1}, ..., a_0\) are constants, and \(n\) is a non-negative integer.

In the given exercise, the functions \(f(x) = 2x + 3\) and \(g(x) = x - 1\) are both polynomial functions of degree 1, also known as linear polynomials.

Key characteristics of polynomial functions include:

• Continuous nature: They are continuous over all real numbers.
• Smooth curves: Their graphs are smooth and unbroken.

When you perform operations such as addition, subtraction, or multiplication on polynomial functions, the result is typically another polynomial. For example:

- The sum \(f(x) + g(x) = 3x + 2\) remains a polynomial.
- The difference \(f(x) - g(x) = x + 4\) and the product \(f(x) \cdot g(x) = 2x^2 + x - 3\) are also polynomials.

This consistency makes polynomial functions highly versatile and widely used in various areas of mathematics and applied fields.

Rational Functions

Rational functions are a significant class of functions in mathematics, defined as the ratio of two polynomials. A general rational function can be expressed as \(R(x) = \frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomial functions. The domain of a rational function consists of all real numbers except those values that make the denominator zero.

In the exercise, the rational function \((f/g)(x) = \frac{2x+3}{x-1}\) is formed by dividing the polynomial \(f(x)\) by \(g(x)\). The critical concept here is understanding where \(g(x)\) becomes zero, as it results in division by zero which is undefined.

Key considerations for rational functions include:

• Non-zero denominator: The denominator \(Q(x)\) must not be zero.
• Discontinuities: Graphically, rational functions can have vertical asymptotes at these critical values.

In this case, since \(g(x) = x - 1\) becomes zero at \(x = 1\), this value is excluded from the domain of \(f/g\). The domain is therefore \((−\infty, 1) \cup (1, \infty)\), including all real numbers except \(x = 1\).

Rational functions are prevalent in various contexts, from simple algebraic expressions to complex analyses in calculus and engineering fields.