Problem 30 - Beginning and Intermediate Algebra

Question

Let \(f(x)=3 x-7\) and \(g(x)=x^{2}-4 x-9 .\) Find each of the following and simplify. $$g(-2)$$

Step-by-Step Solution

Answer

The value of \(g(-2)\) is 3.

Step 1: Identify the g(x) function

We are given the function g(x) as \(g(x) = x^2 - 4x - 9\).

Step 2: Substitute the value of x

Now we need to substitute -2 for x in g(x) function: \(g(-2) = (-2)^2 - 4(-2) - 9\).

Step 3: Simplify the expression

Calculate the value of g(-2): \[ g(-2) = 4 + 8 - 9 \]

Step 4: Find the final value

Now, let's find the final value of g(-2): \[ g(-2) = 3 \] The value of g(-2) is 3.

Key Concepts

Algebraic Expressions

Algebraic expressions are fundamental components in mathematics, consisting of numbers, variables, and operations. They do not have an equal sign unless they are placed within an equation, allowing them to represent real-world situations and mathematical ideas in a concise form. When evaluating algebraic expressions, the primary goal is to transform the expression by replacing the variable(s) with specific numbers, followed by simplification of the subsequent arithmetic operations.

• Variables: Represents unknown values, often denoted by letters such as x, y, or z.
• Constants: Fixed numerical values like 3 or -7.
• Operators: Include addition, subtraction, multiplication, and division.

Working with algebraic expressions provides a strong foundation for solving equations and understanding more complex mathematical functions.

Polynomial Function

A polynomial function is a type of algebraic expression that consists of variables raised to different powers, multiplied by coefficients, and summed together. These functions are characterized by their degree, which is the highest power of the variable in the expression. In the function provided in the exercise, the polynomial function is described as:\[ g(x) = x^2 - 4x - 9 \]

• This is a second-degree polynomial because the highest power of x is 2.
• The coefficients in this function are 1 for \(x^2\), -4 for \(x\), and -9 for the constant term.

Polynomials are versatile and appear frequently in various areas of mathematics and real-world applications, such as physics and economics. Understanding polynomials aids in predictive analysis and provides insights into the behavior of mathematical models.

Substitution Method

The substitution method is a technique used for evaluating algebraic expressions or solving equations by replacing variables with given numbers. This method simplifies the expression to a single numerical result. Let's break down its application in the given exercise:1. **Identify the Function:** First, identify the polynomial function involved, which in this case is \( g(x) = x^2 - 4x - 9 \).2. **Insert the Value:** Substitute the given value, \(-2\), into the polynomial function wherever \( x \) appears: \[ g(-2) = (-2)^2 - 4(-2) - 9 \] 3. **Calculate:** This involves performing arithmetic operations: - **Square:** \((-2)^2\) gives 4. - **Multiply:** \(-4 \times -2\) gives 8. - **Simplify:** Combine all terms to reach the final result: \[ g(-2) = 4 + 8 - 9 = 3 \] This method is powerful for simplifying problems by transforming complex expressions into simple calculations. It is widely used in algebra and calculus to analyze functions' values at specific points.