Problem 46 - College Algebra with Modeling and Visualization

Question

Exercises \(41-52:\) For the given \(g(x),\) evaluate each of the following. $$ \begin{array}{lllll} \text { (a) } g(-3) & \text { (b) } g(b) & \text { (c) } g\left(x^{3}\right) & \text { (d) } g(2 x-3) \end{array} $$ $$ g(x)=2 x^{2}-x-9 $$

Step-by-Step Solution

Answer

(a) 12, (b) \(2b^2 - b - 9\), (c) \(2x^6 - x^3 - 9\), (d) \(8x^2 - 26x + 12\)."

Step 1: Evaluate g(-3)

Plug in -3 for x in the function \(g(x) = 2x^2 - x - 9\). Calculate \(g(-3)\) as follows:\[g(-3) = 2(-3)^2 - (-3) - 9 = 2(9) + 3 - 9 = 18 + 3 - 9 = 12.\] Thus, \(g(-3) = 12.\)

Step 2: Evaluate g(b)

Plug b for x in the function \(g(x) = 2x^2 - x - 9\). Calculate \(g(b)\) as follows: \(g(b) = 2b^2 - b - 9.\)

Step 3: Evaluate g(x^3)

Plug \(x^3\) for x in the function \(g(x) = 2x^2 - x - 9\). Calculate \(g(x^3)\) as follows:\[g(x^3) = 2(x^3)^2 - x^3 - 9 = 2x^6 - x^3 - 9.\]

Step 4: Evaluate g(2x - 3)

Plug \(2x - 3\) for x in the function \(g(x) = 2x^2 - x - 9\). Calculate \(g(2x - 3)\) as follows:\[g(2x - 3) = 2(2x - 3)^2 - (2x - 3) - 9.\] Expand \((2x - 3)^2 = 4x^2 - 12x + 9\). Thus,\[g(2x - 3) = 2(4x^2 - 12x + 9) - 2x + 3 - 9 = 8x^2 - 24x + 18 - 2x + 3 - 9 = 8x^2 - 26x + 12.\]

Key Concepts

Polynomial Functions

Polynomial functions are expressions involving variables raised to whole number powers and coefficients that can be either real or complex numbers. A polynomial is expressed in the form:

• Constant term
• Linear term (\(ax\))
• Quadratic term (\(ax^2\))
• Higher degree terms (\(ax^n\))

For example, the given function \(g(x) = 2x^2 - x - 9\) is a quadratic polynomial because its highest degree is 2. Polynomial functions are used to model a wide range of real-world situations.
They are crucial in understanding relationships between variables, especially in physics and economics. Understanding how to work with polynomials is key to higher-level mathematics.

Substitution Method

The substitution method is a technique used to evaluate functions, solve equations, and simplify expressions. It involves replacing a variable with a specific value or another expression.
In this exercise, we use substitution to find values such as \(g(-3)\) by replacing each instance of \(x\) with \(-3\) in the polynomial \(g(x) = 2x^2 - x - 9\).
When you substitute a value into a polynomial, follow these steps:

• Identify the expression given and understand what needs substitution.
• Replace the variable with the required value or expression.
• Simplify the expression to get an answer.

This method helps in finding the function's value at any given input. Mastering the substitution method is fundamental for solving many mathematical problems.

Algebraic Manipulation

Algebraic manipulation involves rearranging, simplifying, and transforming algebraic expressions to make calculations easier or to solve equations.
In the provided solution, algebraic manipulation is used in steps such as expanding \((2x - 3)^2\) in \(g(2x - 3)\).
Here are some basic algebraic strategies:

• Expanding brackets: Use distributive properties to simplify expressions, e.g., \((a + b)^2 = a^2 + 2ab + b^2\).
• Combining like terms: Add or subtract similar terms, such as \(2x\) and \(-2x\)
• Factorizing expressions: Break down complex terms into simpler multiplicative components.

These manipulations are essential for validating solutions and making sure answers are in their simplest form. They also provide a more intuitive understanding of relationships within the functions.