Problem 23 - Algebra 1

Question

Evaluate the function when \(x=2, x=0,\) and \(x=-3\). $$ g(x)=1.25 x $$

Step-by-Step Solution

Answer

The function \(g(x) = 1.25x\) evaluated at \(x = 2\), \(x = 0\), and \(x = -3\) yields \(g(2) = 2.5\), \(g(0) = 0\), and \(g(-3) = -3.75\), respectively.

Step 1: Substituting \(x = 2\) into the function

Substitute \(x = 2\) into the function \(g(x) = 1.25x\). This gives \(g(2) = 1.25 * 2\).

Step 2: Compute the result for \(x = 2\)

Calculate the result of \(g(2) = 1.25 * 2 = 2.5\).

Step 3: Substituting \(x = 0\) into the function

Substitute \(x = 0\) into the function \(g(x) = 1.25x\). This gives \(g(0) = 1.25 * 0\).

Step 4: Compute the result for \(x = 0\)

Calculate the result of \(g(0) = 1.25 * 0 = 0\).

Step 5: Substituting \(x = -3\) into the function

Substitute \(x = -3\) into the function \(g(x) = 1.25x\). This gives \(g(-3) = 1.25 * -3\).

Step 6: Compute the result for \(x = -3\)

Calculate the result of \(g(-3) = 1.25 * -3 = -3.75\).

Key Concepts

Linear Functions

In mathematics, a linear function is a function that creates a straight line when graphed. Think of a straight road, where the distance you travel is directly proportional to the time you spend walking. Linear functions can be described using the formula \( f(x) = mx + b \), where:

• \( m \) is the slope, which measures the function's steepness or incline.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

In our exercise, the function \( g(x) = 1.25x \) is a special type of linear function called a direct variation. This means it has no y-intercept (\( b = 0 \)), making the line pass through the origin at (0,0). The slope is 1.25, indicating that for every unit increase in \( x \), \( g(x) \) increases by 1.25. Linear functions like \( g(x) = 1.25x \) are foundational in algebra and provide a basic understanding of how changes in one variable affect another. The simplicity of linear functions helps build your confidence in evaluating and graphing more complex functions later on.

Substitution

Substitution is a fundamental technique in mathematics that allows us to replace a variable with a numerical value or another expression to solve equations. In the given exercise, substitution involves finding the value of the function \( g(x) \) for different values of \( x \). The process of substitution can be broken down into simple steps:

• Identify the variable to be replaced (in this case, \( x \)).
• Know the specific values or expressions replacing the variable (here we have \( x = 2, 0, -3 \)).
• Substitute the values into the function \( g(x) = 1.25x \), replacing every \( x \) with the specified number.

For example, when substituting \( x = 2 \), we replace \( x \) in the equation, resulting in \( g(2) = 1.25 \times 2 \). By systematically applying substitution, complex problems become more manageable as you're working with specific numbers instead of abstract symbols.

Basic Arithmetic

Basic arithmetic is the cornerstone of math, involving the simple operations of addition, subtraction, multiplication, and division. In our function evaluation exercise, we primarily focus on multiplication, one of these basic operations. Let's look at why mastering basic arithmetic, particularly multiplication, is crucial:

• It enables us to quickly solve problems, like calculating the result of \( g(x) = 1.25x \) for different \( x \) values.
• Understanding arithmetic helps us transition to more advanced mathematics, where quick and accurate calculations are essential.

This exercise asks for such arithmetic operations at different substitution points (\( x = 2, 0, -3 \)). For instance:

• At \( x = 2 \): You calculate \( 1.25 \times 2 = 2.5 \).
• At \( x = 0 \): You find \( 1.25 \times 0 = 0 \), showing the effect of multiplying by zero.
• At \( x = -3 \): You determine \( 1.25 \times (-3) = -3.75 \), illustrating multiplication with a negative number results in a negative product.

Grasping these basic arithmetic skills is vital as they form the basis for understanding more intricate math concepts that you will encounter as you progress in your studies.