Finding The Difference Between Functions S(x) And R(x) A Step-by-Step Guide

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In the realm of mathematics, functions serve as fundamental building blocks for modeling and understanding relationships between variables. When dealing with multiple functions, operations like addition, subtraction, multiplication, and division can be performed to create new functions with unique characteristics. In this comprehensive guide, we will delve into the concept of finding the difference between two functions, specifically focusing on the functions r(x) = -x² + 3x and s(x) = 2x + 1. Our goal is to determine the resulting function (s - r)(x), providing a step-by-step explanation and illuminating the underlying principles.

Understanding Function Subtraction

At its core, function subtraction involves finding the difference between the outputs of two functions for a given input value. In simpler terms, if we have two functions, f(x) and g(x), their difference, denoted as (f - g)(x), is obtained by subtracting the value of g(x) from the value of f(x) for the same input x. Mathematically, this can be expressed as:

(f - g)(x) = f(x) - g(x)

The result of this operation is a new function that represents the pointwise difference between the original functions. This concept is crucial in various mathematical applications, including calculus, where it is used to find the rate of change between two functions, and in real-world scenarios, where it can model the difference in costs, profits, or other relevant quantities.

Applying Function Subtraction to r(x) and s(x)

Now, let's apply the concept of function subtraction to the given functions:

r(x) = -x² + 3x s(x) = 2x + 1

To find (s - r)(x), we need to subtract r(x) from s(x):

(s - r)(x) = s(x) - r(x)

Substituting the expressions for s(x) and r(x), we get:

(s - r)(x) = (2x + 1) - (-x² + 3x)

To simplify this expression, we need to distribute the negative sign to the terms inside the parentheses:

(s - r)(x) = 2x + 1 + x² - 3x

Now, we can combine like terms to obtain the final expression for (s - r)(x):

(s - r)(x) = x² - x + 1

Therefore, the difference between the functions s(x) and r(x) is the quadratic function (s - r)(x) = x² - x + 1. This new function represents the vertical distance between the graphs of s(x) and r(x) at any given point x. Understanding how to perform function subtraction is essential for various mathematical and real-world applications, allowing us to analyze the relationships and differences between functions.

Step-by-Step Calculation of (s - r)(x)

To solidify our understanding, let's break down the calculation of (s - r)(x) into a step-by-step process:

Step 1: Write down the expressions for the functions s(x) and r(x).

We are given:

s(x) = 2x + 1 r(x) = -x² + 3x

This initial step is crucial for organizing the information and ensuring that we have a clear understanding of the functions we are working with. By explicitly stating the expressions, we minimize the chances of errors in subsequent calculations.

Step 2: Express (s - r)(x) as s(x) - r(x).

This step clarifies the operation we need to perform. We are finding the difference between two functions, which means we need to subtract the expression for r(x) from the expression for s(x):

(s - r)(x) = s(x) - r(x)

This step is important because it sets the stage for the substitution process in the next step.

Step 3: Substitute the expressions for s(x) and r(x) into the equation.

Now, we replace s(x) and r(x) with their respective expressions:

(s - r)(x) = (2x + 1) - (-x² + 3x)

This substitution is a key step in the process, as it allows us to work with the algebraic expressions that define the functions. It's important to pay close attention to the signs when substituting, especially when dealing with negative signs.

Step 4: Distribute the negative sign to the terms inside the parentheses.

To remove the parentheses, we need to distribute the negative sign to each term inside the second set of parentheses:

(s - r)(x) = 2x + 1 + x² - 3x

This step is crucial for simplifying the expression. Failing to distribute the negative sign correctly can lead to errors in the final result.

Step 5: Combine like terms to simplify the expression.

Now, we identify and combine terms with the same variable and exponent:

(s - r)(x) = x² + (2x - 3x) + 1

(s - r)(x) = x² - x + 1

This final step involves simplifying the expression by combining like terms. This makes the result more concise and easier to interpret. The simplified expression represents the difference between the two functions, and it can be used to analyze the relationship between them.

By following these steps carefully, we arrive at the difference function (s - r)(x) = x² - x + 1. This step-by-step approach not only provides the correct answer but also reinforces the underlying concepts of function subtraction and algebraic manipulation. Understanding these steps is essential for tackling more complex problems involving function operations.

Visualizing the Difference: Graphing s(x), r(x), and (s - r)(x)

While algebraic manipulation provides a precise way to find the difference between functions, visualizing the graphs of the functions offers an intuitive understanding of their relationship. By plotting s(x), r(x), and (s - r)(x) on the same coordinate plane, we can observe how the difference function represents the vertical distance between the original functions.

Graphing s(x) = 2x + 1

The function s(x) = 2x + 1 is a linear function, which means its graph is a straight line. To plot this line, we can find two points on the line and connect them. For example:

  • When x = 0, s(0) = 2(0) + 1 = 1. So, the point (0, 1) lies on the line.
  • When x = 1, s(1) = 2(1) + 1 = 3. So, the point (1, 3) lies on the line.

By plotting these points and drawing a line through them, we obtain the graph of s(x). The slope of this line is 2, which indicates the rate of change of the function, and the y-intercept is 1, which is the value of the function when x = 0.

Graphing r(x) = -x² + 3x

The function r(x) = -x² + 3x is a quadratic function, which means its graph is a parabola. The negative coefficient of the term indicates that the parabola opens downwards. To plot this parabola, we can find its vertex, x-intercepts, and a few additional points.

  • Vertex: The x-coordinate of the vertex can be found using the formula x = -b / 2a, where a = -1 and b = 3. So, x = -3 / (2 * -1) = 1.5. The y-coordinate of the vertex is r(1.5) = -(1.5)² + 3(1.5) = 2.25. Therefore, the vertex is at (1.5, 2.25).

  • X-intercepts: To find the x-intercepts, we set r(x) = 0 and solve for x:

    -x² + 3x = 0

    x(-x + 3) = 0

    So, x = 0 and x = 3 are the x-intercepts.

By plotting the vertex, x-intercepts, and a few additional points, we can sketch the graph of r(x). The parabola opens downwards, with its highest point at the vertex.

Graphing (s - r)(x) = x² - x + 1

The function (s - r)(x) = x² - x + 1 is also a quadratic function, but this time, the coefficient of the term is positive, indicating that the parabola opens upwards. To plot this parabola, we can find its vertex and a few additional points.

  • Vertex: The x-coordinate of the vertex can be found using the formula x = -b / 2a, where a = 1 and b = -1. So, x = -(-1) / (2 * 1) = 0.5. The y-coordinate of the vertex is (s - r)(0.5) = (0.5)² - 0.5 + 1 = 0.75. Therefore, the vertex is at (0.5, 0.75).

By plotting the vertex and a few additional points, we can sketch the graph of (s - r)(x). The parabola opens upwards, with its lowest point at the vertex.

Interpreting the Graphs

By plotting all three functions on the same graph, we can visualize the difference between s(x) and r(x). The graph of (s - r)(x) represents the vertical distance between the graphs of s(x) and r(x) at any given point x. Where s(x) is above r(x), the value of (s - r)(x) is positive, and where r(x) is above s(x), the value of (s - r)(x) is negative. The points where the graphs of s(x) and r(x) intersect correspond to the points where (s - r)(x) = 0.

Visualizing the graphs of functions and their differences provides a powerful tool for understanding their relationships and behaviors. It allows us to connect the algebraic expressions with their geometric representations, leading to a deeper understanding of mathematical concepts.

Real-World Applications of Function Subtraction

Function subtraction is not merely an abstract mathematical concept; it has numerous applications in various real-world scenarios. By understanding how to find the difference between functions, we can model and analyze real-world situations involving comparisons, changes, and optimization. Let's explore some practical examples:

1. Profit Analysis

In business, profit is calculated as the difference between revenue and cost. If we have functions representing the revenue, R(x), and the cost, C(x), of producing x units of a product, then the profit function, P(x), can be expressed as:

P(x) = R(x) - C(x)

By subtracting the cost function from the revenue function, we obtain a new function that represents the profit generated at different production levels. This allows businesses to analyze their profitability, identify break-even points, and optimize production to maximize profits.

2. Temperature Differences

In meteorology and climate science, understanding temperature differences is crucial for weather forecasting and climate modeling. If we have functions representing the temperature at two different locations or at different times, we can use function subtraction to find the temperature difference.

For example, if T₁(t) represents the temperature at location 1 at time t, and T₂(t) represents the temperature at location 2 at the same time, then the temperature difference function, ΔT(t), can be expressed as:

ΔT(t) = T₁(t) - T₂(t)

This function allows us to analyze temperature gradients, track temperature changes over time, and understand the factors driving these changes.

3. Distance and Displacement

In physics and engineering, function subtraction can be used to analyze the motion of objects. If we have functions representing the position of an object at different times, we can use function subtraction to find the displacement of the object over a given time interval.

For example, if s₁(t) represents the position of an object at time t₁, and s₂(t) represents the position of the object at time t₂, then the displacement of the object between t₁ and t₂ can be expressed as:

Δs = s₂(t₂) - s₁(t₁)

This calculation is fundamental in understanding the motion of objects and is used in various applications, including trajectory analysis, robotics, and control systems.

4. Comparing Growth Rates

In biology and economics, function subtraction can be used to compare the growth rates of different populations or investments. If we have functions representing the size of two populations or the value of two investments over time, we can use function subtraction to find the difference in their growth.

For example, if P₁(t) represents the population of species 1 at time t, and P₂(t) represents the population of species 2 at the same time, then the difference in population growth can be expressed as:

ΔP(t) = P₁(t) - P₂(t)

By analyzing this function, we can understand which population is growing faster and the factors driving these differences.

These are just a few examples of the many real-world applications of function subtraction. By understanding this concept, we can model and analyze a wide range of phenomena involving comparisons, changes, and optimization.

Conclusion: Mastering Function Subtraction

In this comprehensive guide, we have explored the concept of function subtraction, providing a step-by-step explanation of how to find the difference between two functions. We specifically focused on the functions r(x) = -x² + 3x and s(x) = 2x + 1, demonstrating how to calculate (s - r)(x) algebraically and visualize the result graphically. We also highlighted the numerous real-world applications of function subtraction, showcasing its importance in various fields.

By mastering the techniques and principles outlined in this guide, you can confidently tackle problems involving function subtraction and apply this knowledge to solve real-world problems. Function subtraction is a fundamental tool in mathematics and its applications, and a solid understanding of this concept will undoubtedly enhance your problem-solving abilities.

Remember, practice is key to mastering any mathematical concept. Work through additional examples and explore different scenarios to solidify your understanding of function subtraction. With dedication and practice, you will be able to confidently apply this concept to solve a wide range of problems.

Keywords: function subtraction, difference of functions, r(x) = -x² + 3x, s(x) = 2x + 1, (s - r)(x), step-by-step calculation, graphing functions, real-world applications, profit analysis, temperature differences, distance and displacement, comparing growth rates.