Problem 51 - 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. $$ -f(x)+4 g(x) $$

Step-by-Step Solution

Answer

The result of \(-f(x) + 4g(x)\) operation is \(-3x^2 + 2x +16\) and the domain is all real numbers (\(x \in (-\infty, +\infty)\)).

Step 1: Define the functions

The functions are defined as follows: \(f(x) = 3x^2+2x-8\) and \(g(x) = x+2.\)

Step 2: Set up the operation

We are asked to calculate the result of \(-f(x) + 4g(x)\). So, for this, we need to make this calculation by first substituting the given functions into this operation: \(-f(x) + 4g(x) = - (3x^2+2x-8) + 4(x+2).\)

Step 3: Simplify the expression

When we simplify the expression, we get: \(- (3x^2+2x-8) + 4(x+2) = -3x^2-2x+8 + 4x+8 = -3x^2 + 2x +16.\)

Step 4: Identify the domain

The simplified expression is a polynomial function, which is defined for all real numbers, so the domain is \(x \in (-\infty, +\infty)\).

Key Concepts

Polynomial Functions

Polynomial functions are a fundamental concept in algebra and calculus. They involve expressions made up of variables raised to whole-number exponents, with each term consisting of a coefficient and one or more such variables. For example, the equation \(f(x) = 3x^2 + 2x - 8\) is a polynomial function of degree 2 because its highest exponent is 2.

• Polynomial functions can have one or more terms. Each term is a product of a constant and a variable raised to a non-negative integer exponent.
• The degree of a polynomial is determined by the highest exponent of its variables.
• Polynomials are smooth, continuous functions, making them simple to work with when graphing.

Understanding polynomial functions is vital as they form the building blocks for more complex mathematical concepts. They appear in a variety of disciplines, from physics to economics, making them a versatile tool in mathematical modeling.

Domain of a Function

The domain of a function refers to all the possible inputs (or \(x\)-values) for which the function is defined. Knowing how to determine the domain is essential when working with functions because it tells us where the function is valid and usable.

• For polynomial functions, the domain is generally all real numbers, \(x \in (-\infty, +\infty)\), because polynomials are defined for every real number.
• Other types of functions, such as rational functions, may have restricted domains due to undefined values (like division by zero).

Identifying the domain of a function is a crucial first step before performing any operations, ensuring that any mathematical manipulations are valid throughout the range of inputs.

Simplifying Expressions

Simplifying expressions is the process of rewriting a mathematical expression in its simplest form. This often involves applying algebraic rules to combine like terms, reduce fractions, or factor expressions whenever possible.

• To simplify any given algebraic expression, begin by performing arithmetic operations such as addition, subtraction, multiplication, and division.
• Combine like terms, which are terms that contain the same variables raised to the same powers, to consolidate the expression.
• Pay attention to signs, especially when distributing a negative sign over an expression, as seen in the calculation \(-f(x) + 4g(x) = -3x^2 + 2x + 16\).

Simplifying makes expressions easier to read and work with, as well as helps in solving equations efficiently. It's a skill applicable across all fields of mathematics, ensuring clarity and accuracy in calculations.