Problem 73 - Elementary and Intermediate Algebra

Question

Let \(f(x)=x^{2}-1\) and \(g(x)=x^{2}-4 .\) Find each function and give its domain. $$ f-g $$

Step-by-Step Solution

Answer

The function \(f-g(x) = 3\) has a domain of all real numbers \(\mathbb{R}\).

Step 1: Identify Function Expressions

First, let's identify the expressions for each function involved in the subtraction. We have \(f(x) = x^2 - 1\) and \(g(x) = x^2 - 4\).

Step 2: Subtract the Functions

The subtraction of the two functions \(f(x)\) and \(g(x)\) is expressed as \((f-g)(x) = f(x) - g(x)\). Plugging in the expressions, we get \((f-g)(x) = (x^2 - 1) - (x^2 - 4)\).

Step 3: Simplify the Expression

We now simplify the expression from the previous step:\[(f-g)(x) = (x^2 - 1) - (x^2 - 4) = x^2 - 1 - x^2 + 4\]. The \(x^2\) terms cancel each other out, resulting in \((f-g)(x) = 3\).

Step 4: Determine the Domain

The domain of a function is the set of all possible input values \(x\) for which the function is defined. Both \(f(x)\) and \(g(x)\) are polynomials, which are defined for all real numbers. Therefore, the domain of \((f-g)(x)\) is all real numbers \(\mathbb{R}\).

Key Concepts

Functions

Functions are mathematical expressions that describe a relationship between input and output values. They can be thought of as machines that take in a number (input) and produce another number (output). For example, in the function \(f(x) = x^2 - 1\), \(f\) is the name of the function, \(x\) is the input variable (often real numbers), and \(x^2 - 1\) is the output expression. Functions can be expressed in various forms such as equations, tables, or graphs. They are essential tools in both pure and applied mathematics, allowing us to model real-world scenarios. When dealing with functions, operations such as addition, subtraction, multiplication, and division can be performed. To subtract two functions, like \(f(x)\) and \(g(x)\), you essentially subtract their expressions. In our exercise, \((f-g)(x) = f(x) - g(x) = (x^2 - 1) - (x^2 - 4)\). This operation helps to find relationships and differences between two quantities or conditions modeled by the functions.

Domain of a Function

The domain of a function refers to the complete set of possible input values (\(x\)-values) for which the function is defined. For most basic functions, such as polynomial functions, the domain is often all real numbers, symbolized by \(\mathbb{R}\).Understanding the domain is critical because it sets the boundaries for where the function can operate. It helps us know whether plugging in certain values will yield valid output or if special conditions apply (like division by zero, which is undefined). In the case of our functions \(f(x) = x^2 - 1\) and \(g(x) = x^2 - 4\), both are polynomials. Polynomials are defined for all real numbers without restriction. Hence, the domain for \(f(x) - g(x)\) is also all real numbers, \(\mathbb{R}\). This means you can substitute any real number into the function, and you'll get a valid result.

Polynomials

Polynomials are algebraic expressions made up of variables and coefficients, utilizing only the operations of addition, subtraction, multiplication, and non-negative integer exponents. A simple example of a polynomial is \(x^2 - 1\). In general, polynomials can be represented as \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where each \(a_i\) is a constant coefficient, and the highest power of \(x\) (with a nonzero coefficient) dictates the degree of the polynomial.The polynomials \(f(x) = x^2 - 1\) and \(g(x) = x^2 - 4\) from our exercise are quadratic polynomials, meaning their highest degree is 2. Polynomials like these are continuous and differentiable for all real numbers, making them very versatile in mathematical modeling. Subtraction of polynomials involves subtracting corresponding coefficients, as seen in \((f-g)(x) = (x^2 - 1) - (x^2 - 4)\), which simplifies to a constant function \(3\): - The \(x^2\) terms cancel out, leaving the constant result. Polynomials are straightforward yet powerful in understanding and interpreting mathematical relationships and changes in various contexts.