Problem 52 - Algebra 2

Question

Let \(f(x)=3 x^{2}+2 x-8\) and \(g(x)=x+2 .\) Perform each function operation and then find the domain. $$ f(x)-2 g(x) $$

Step-by-Step Solution

Answer

The result of the operation \(f(x) - 2g(x)\) is the function \(3x^{2} - 12\), and the domain of this function is \(-\infty < x <\infty\).

Step 1: Subtract Functions

Start by subtracting \(2g(x)\) from \(f(x)\). Let's write down the functions: \[f(x)=3x^{2}+2x-8\] and \[g(x)=x+2\] Substituting \(g(x)\) in the operation gives \(f(x)-2g(x) = f(x) - 2(x+2)\). Now, we can distribute the 2 into the parentheses: \(f(x) - 2x - 4\). Substitute \(f(x)\) back into the equation thus obtaining \(3x^{2} + 2x -8 -2x -4 = 3x^{2}-8-4\).

Step 2: Simplify the Function

Simplify by collecting like terms. \(3x^{2} - 12 = 3x^{2} -12\). This equation is the simplified function of \(f(x)-2g(x)\).

Step 3: Find the Domain

To find the domain, we need to know the set of all possible x values that can make the function defined. Since our simplified function is a polynomial and there are no denominators (which could equal zero), no radicals (that need positive numbers), and no Logarithms, the domain would be all real numbers. Hence, the domain of \(f(x)-2g(x) = (-\infty, \infty)\).

Key Concepts

Domain of a Function

The concept of the domain of a function refers to all the possible input values (x-values) that will produce a valid output when plugged into the function. In simpler terms, the domain includes every number you can use for the variable x without causing any mathematical problems, like division by zero or the square root of a negative number.

For our situation with the problem statement, we are dealing with a polynomial resulting from the operation of subtraction between two functions: \(f(x)-2g(x)\). Since polynomial functions, like the one in this exercise, do not have any restrictions—as they don’t involve denominators, square roots, or logarithms—their domain is typically all real numbers.

To express this in interval notation for the given exercise, we write the domain as \((-\infty, \infty)\). This covers every possible number from negative infinity to positive infinity, signifying that there are no constraints on the values x can take within this function.

Polynomial Functions

Polynomial functions are an essential part of algebra and are widely used across various fields of mathematics and science. A polynomial is an expression consisting of variables, exponents, and coefficients. The general form is \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\), where each \(a\) represents a coefficient and \(n\) is a non-negative integer indicator of the degree of the polynomial.

In the provided exercise, \(f(x) = 3x^2 + 2x - 8 \) is a polynomial function. It is specifically a quadratic polynomial because the highest power of x is 2. Polynomials can be plotted as smooth and continuous graphs, without any breaks. This interrelatedness implies that polynomials are straightforward regarding the domain and range, as the given polynomial allows any real number for x.

Moreover, polynomial functions are closed under addition, subtraction, and multiplication, meaning the result of these operations is still a polynomial. This is important as it confirms why the operation with \(f(x) - 2g(x)\) results in another polynomial: \(3x^2 - 12\).

Function Simplification

Function simplification involves reducing a mathematical expression or equation to its simplest form. The aim is to make the function as straightforward as possible while retaining its value and properties.

For instance, in the function operation \(f(x) - 2g(x)\), the first step is to apply the operation: replace \(g(x)\) with its expression and distribute any coefficients. This step requires careful handling of the arithmetic operations involved to avoid errors.

Next, combining like terms is critical. Like terms are those that have the same variables and powers. In this problem, after substituting and expanding we are left with \(3x^2 - 8 - 2x - 4\). By combining like terms, the expression simplifies to \(3x^2 - 12\).

Simplification allows functions to be evaluated more efficiently, making it easier to see the core structure of the expression, leading into a better understanding and analysis of the function properties such as domain or range.