Problem 41 - College Algebra with Modeling and Visualization

Question

Exercises \(41-52:\) For the given \(g(x),\) evaluate each of the following. $$ \begin{array}{lllll} \text { (a) } g(-3) & \text { (b) } g(b) & \text { (c) } g\left(x^{3}\right) & \text { (d) } g(2 x-3) \end{array} $$ $$ g(x)=2 x+1 $$

Step-by-Step Solution

Answer

(a) -5, (b) 2b+1, (c) 2x^3+1, (d) 4x-5.

Step 1: Evaluate g(-3)

To find \( g(-3) \), substitute \( -3 \) into the function \( g(x) = 2x + 1 \). Calculate: \( g(-3) = 2(-3) + 1 \). Simplifying, \( g(-3) = -6 + 1 = -5 \).

Step 2: Evaluate g(b)

To find \( g(b) \), substitute \( b \) into the function \( g(x) = 2x + 1 \). Calculate: \( g(b) = 2b + 1 \). Simplifying, the function remains: \( g(b) = 2b + 1 \).

Step 3: Evaluate g(x^3)

To find \( g(x^3) \), substitute \( x^3 \) into the function \( g(x) = 2x + 1 \). Calculate: \( g(x^3) = 2(x^3) + 1 \). Simplifying, \( g(x^3) = 2x^3 + 1 \).

Step 4: Evaluate g(2x-3)

To find \( g(2x-3) \), substitute \( 2x-3 \) into the function \( g(x) = 2x + 1 \). Calculate: \( g(2x-3) = 2(2x-3) + 1 \). Simplify the expression: \( g(2x-3) = 4x - 6 + 1 = 4x - 5 \).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They form the basis of algebra and are used to represent mathematical relationships in a compact form.
For example, when you see something like \(2x + 1\), you're looking at an algebraic expression. This particular expression means that whatever value \(x\) takes, you multiply it by 2 and then add 1.
Algebraic expressions can be manipulated through addition, subtraction, multiplication, and division. It is important to remember:

• Variables can represent different values.
• Operations follow the order of precedence (parentheses, exponents, multiplication and division, addition and subtraction).
• Expressions can be simplified by combining like terms or factoring.

Understanding algebraic expressions is key to evaluating functions and solving equations. They are foundational elements you'll encounter throughout all of algebra.

Substitution Method

The substitution method is a technique used to simplify expressions or solve equations by replacing a variable with a given number or expression. This method helps in evaluating functions efficiently.
Let's take the example from the exercise: If you want to find \(g(-3)\) for the function \(g(x) = 2x + 1\), you substitute \(-3\) for \(x\) in the expression:

• Original function: \(g(x) = 2x + 1\)
• Substitute \(-3\): \(g(-3) = 2(-3) + 1\)
• Calculate: \(g(-3) = -6 + 1 = -5\)

The same method is applied if you substitute a variable, like \(b\), or more complex expressions, like \(x^3\) or \(2x-3\). It's essential to follow every calculation step carefully. This ensures that substitution is done correctly, yielding accurate results.

Function Notation

Function notation is a way of writing functions denoting the relationship between two variables, usually written as \(f(x)\), where \(f\) is the function name, and \(x\) is the variable.
This notation helps in understanding which input variable you're dealing with and what output the function provides. It's like a label that tells you, "This operation is performed on this input."
In the context of the exercise, \(g(x) = 2x + 1\) translates to: "The function \(g\) takes \(x\) and outputs \(2x + 1\)." Here's why function notation is useful:

• It clearly represents the relationship between varying quantities.
• Makes it easy to substitute different values into the function.
• Facilitates communication in math by providing a concise representation of functional relationships.

When you see \(g(-3)\), \(g(b)\), or \(g(2x-3)\), it tells you exactly what is being input into the function and what operation will be performed.