Problem 66 - Elementary and Intermediate Algebra

Question

Let \(f(x)=3 x-2\) and \(g(x)=2 x^{2}+1 .\) Find each function and give its domain. $$ f+g $$

Step-by-Step Solution

Answer

The function \\(f+g = 2x^2 + 3x - 1\\) with domain all real numbers.

Step 1: Understanding the Problem

We need to find the expression for the function \(f+g\) and also determine its domain. We have two functions defined as \(f(x) = 3x - 2\) and \(g(x) = 2x^2 + 1\).

Step 2: Adding the Functions

To find the function \(f+g\), we add \(f(x)\) and \(g(x)\). This gives us \((f+g)(x) = f(x) + g(x) = (3x - 2) + (2x^2 + 1)\).

Step 3: Simplifying the Sum

Simplify the expression obtained by adding the two functions: \(2x^2 + 3x + 1 - 2 = 2x^2 + 3x - 1\). Thus, \( (f+g)(x) = 2x^2 + 3x - 1 \).

Step 4: Determining the Domain

Both \(f(x) = 3x - 2\) and \(g(x) = 2x^2 + 1\) are polynomial functions. The domain of any polynomial function is all real numbers. Therefore, the domain of \(f+g\) is also all real numbers.

Key Concepts

Polynomial Functions

Polynomial functions are expressions involving variables raised to whole number powers. They consist of terms that are the product of a constant coefficient and a variable, each raised to a non-negative integer exponent. For example, both functions in our exercise, \( f(x) = 3x - 2 \) and \( g(x) = 2x^2 + 1 \), are polynomial functions.

Here are key features of polynomial functions:

• Degree: The largest exponent of the variable in a polynomial determines its degree. For instance, \( f(x) = 3x - 2 \) is a linear polynomial (degree 1), while \( g(x) = 2x^2 + 1 \) is quadratic (degree 2).
• Coefficients: These are the constants multiplying the variable terms, such as 3 in \( 3x \) and 2 in \( 2x^2 \).
• Constant term: Any term without a variable, such as -2 in \( 3x - 2 \) or 1 in \( 2x^2 + 1 \).

Polynomial functions are smooth and continuous, meaning they don't have breaks, holes, or sharp corners.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. When dealing with polynomial functions, the domain is typically very straightforward. Because these functions are continuous and defined for every real number, they have no restrictions like roots or denominators that could otherwise pose limitations.

For our specific functions \( f(x) = 3x - 2 \) and \( g(x) = 2x^2 + 1 \), the domain is all real numbers \((\mathbb{R})\). That is, you can plug any real number into these functions and get a valid output. Adding these functions as in \( f+g \) doesn't change this property, meaning even the sum function \((f+g)(x)\) = \( 2x^2 + 3x - 1 \) also has a domain of all real numbers.

To put it plainly: if you can think of a number, you can plug it into these functions without any problem!

Real Numbers

Real numbers include everything you can think of on the number line - from negative numbers, like -3 or -5.5, to zero, and positive numbers, like 7 or 2.4. They are called 'real' because they don't include imaginary numbers, which are another concept entirely.

Some common characteristics of real numbers include:

• Rational Numbers: These are numbers that can be expressed as fractions, like \( \frac{1}{2} \) or \( -\frac{3}{4} \).
• Irrational Numbers: Numbers that can't be written as a simple fraction. Famous examples include \(\pi\) (Pi) and \(\sqrt{2}\).
• Integers: Whole numbers without fractions, extending from negative to positive. For example, -1, 0, and 5.

When we say the domain of a polynomial function is all real numbers, we're saying you can input any kind of real number into the function, and it will work just fine.