Problem 43 - College Algebra with Modeling and Visualization

Question

Exercises \(41-52:\) For the given \(g(x),\) evaluate each of the following. $$ \begin{array}{lllll} \text { (a) } g(-3) & \text { (b) } g(b) & \text { (c) } g\left(x^{3}\right) & \text { (d) } g(2 x-3) \end{array} $$ $$ g(x)=2(x+3)^{2}-4 $$

Step-by-Step Solution

Answer

(a) -4, (b) \(2(b+3)^2-4\), (c) \(2((x^3)+3)^2-4\), (d) \(8x^2-4\)."

Step 1: Evaluate g(-3)

To find \( g(-3) \), substitute \( -3 \) for \( x \) in the function \( g(x) = 2(x+3)^2 - 4 \). Calculate as follows:1. Substitute: \( g(-3) = 2((-3)+3)^2 - 4 \)2. Simplify inside the parentheses: \( g(-3) = 2(0)^2 - 4 \)3. Calculate: \( g(-3) = 2 imes 0 - 4 \)4. Result: \( g(-3) = -4 \).So, \( g(-3) = -4 \).

Step 2: Evaluate g(b)

To find \( g(b) \), substitute \( b \) for \( x \) in the function \( g(x) = 2(x+3)^2 - 4 \). Perform the following steps:1. Substitute: \( g(b) = 2(b+3)^2 - 4 \).Thus, \( g(b) = 2(b+3)^2 - 4 \) is the expression for \( g(b) \).

Step 3: Evaluate g(x^3)

To find \( g(x^3) \), substitute \( x^3 \) for \( x \) in the function \( g(x) = 2(x+3)^2 - 4 \). Compute as follows:1. Substitute: \( g(x^3) = 2((x^3)+3)^2 - 4 \).Thus, \( g(x^3) = 2((x^3)+3)^2 - 4 \) is the expression for \( g(x^3) \).

Step 4: Evaluate g(2x-3)

To find \( g(2x-3) \), substitute \( 2x-3 \) for \( x \) in the function \( g(x) = 2(x+3)^2 - 4 \). Calculate the expression:1. Substitute: \( g(2x-3) = 2((2x-3)+3)^2 - 4 \).2. Simplify inside: \( g(2x-3) = 2(2x)^2 - 4 \).3. Expand: \( g(2x-3) = 2(4x^2) - 4 \).4. Final result: \( g(2x-3) = 8x^2 - 4 \).Therefore, \( g(2x-3) = 8x^2 - 4 \).

Key Concepts

Algebraic Functions

Algebraic functions are equations made up of polynomial expressions. They involve variables, numbers, and operations like addition, subtraction, multiplication, and raising to a power. These functions form the backbone of many algebraic calculations and are crucial for solving equations within mathematics. In the exercise, the function given is \[ g(x) = 2(x+3)^2 - 4 \]This means that for any input value of \( x \), you need to replace \( x \) in the equation and simplify.

• The function is made up of a polynomial — in this case, \((x+3)^2\), which is then multiplied by 2 and reduced by 4.
• Understanding algebraic functions helps you evaluate the function for different inputs.
• Recognizing the structure and operations within will help you know how to substitute and simplify.

Algebraic functions like these can model a variety of real-life situations, and learning to manipulate them provides insight into many practical problems.

Substitution Method

The substitution method is a technique used in algebra to evaluate functions. It involves replacing the variable(s) in a function with the specified value(s) or expression(s). This technique is fundamental for calculating the output of algebraic functions for specific inputs. Let's explore how it works:- When asked to evaluate \( g(-3) \), substitution places \(-3\) for \( x \) in the function \( g(x) = 2(x+3)^2 - 4 \).- This results in: \ g(-3) = 2((-3)+3)^2 - 4 \ which simplifies to \(-4\).- For \( g(b) \), substitute \( b \) for \( x \): \[ g(b) = 2(b+3)^2 - 4 \]The substitution method also applies to more complex expressions like \( g(x^3) \) and \( g(2x-3) \). Simply rewrite the function with the substituted input and simplify as needed:- For example, \( g(2x-3) \) becomes: \ g(2x-3) = 2((2x-3)+3)^2 - 4 \ \[ \text{and simplifies to } \; g(2x-3) = 8x^2 - 4 \]

Polynomial Expressions

Polynomial expressions are algebraic expressions composed exclusively of terms in the form \( ax^n \) where \( a \) is a constant, \( n \) is a non-negative integer, and \( x \) is the variable. In our function \( g(x) = 2(x+3)^2 - 4 \), the polynomial part is \((x+3)^2\):- Here, \( x+3 \) gets squared, expanding it to a polynomial.- Polynomials like \((x+3)^2\) simplify to \( x^2 + 6x + 9 \).- Multiplying by 2 results in \( 2x^2 + 12x + 18 \), then adjust with \(-4\) giving us the final expression inside the function.Understanding polynomial expressions in functions is all about recognizing and managing terms:

• Each term in the expression plays a specific role based on its degree, which is the value of \( n \).
• In algebra, operations with polynomials include adding, subtracting, multiplying, and, occasionally, dividing.

Being adept with polynomial expressions allows you to efficiently expand and simplify functions in algebra. This is vital for evaluating algebraic functions accurately.