Section 3 Topic 3 Adding And Subtracting Functions

Adding and subtracting functions is a fundamental concept in mathematics that expands the operations we perform on numbers to operations on functions. This allows us to combine functions in meaningful ways, creating new functions with unique properties and behaviors. Mastering this concept is crucial for understanding more advanced topics in calculus, differential equations, and other areas of mathematics.

Introduction

When we talk about adding and subtracting functions, we are essentially combining two or more functions to create a new function. This new function's output at any given point is determined by the sum or difference of the outputs of the original functions at that same point. This is a powerful tool for modeling real-world phenomena, such as combining cost functions or analyzing the net effect of opposing forces. Understanding how to perform these operations, along with their notations and properties, is key to success in higher-level mathematics.

Notation and Definitions

Let's define two functions, f(x) and g(x). The sum of these two functions, denoted as (f + g)(x), is defined as:

(f + g)(x) = f(x) + g(x)

Similarly, the difference of these two functions, denoted as (f - g)(x), is defined as:

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

These definitions might seem straightforward, but they are the foundation for more complex operations. The domain of the resulting function (either the sum or the difference) is the intersection of the domains of the original functions, f(x) and g(x). This is because the resulting function is only defined where both f(x) and g(x) are defined.

Steps to Add and Subtract Functions

Adding and subtracting functions involves a series of straightforward steps:

1. Identify the functions: Clearly identify the functions f(x) and g(x) that you want to add or subtract.
2. Determine the operation: Decide whether you need to add (f + g)(x) or subtract (f - g)(x).
3. Perform the operation: Add or subtract the functions as defined:
   • For addition: (f + g)(x) = f(x) + g(x)
   • For subtraction: (f - g)(x) = f(x) - g(x)
4. Simplify the expression: Simplify the resulting expression by combining like terms.
5. Determine the domain: Find the domain of the resulting function, which is the intersection of the domains of f(x) and g(x).

Examples of Adding and Subtracting Functions

Let's work through a few examples to illustrate these steps.

Example 1: Adding Polynomial Functions

Suppose we have two functions:

• f(x) = 3x^2 + 2x - 1
• g(x) = x^2 - 4x + 5

We want to find (f + g)(x).

1. Identify the functions: f(x) = 3x^2 + 2x - 1 and g(x) = x^2 - 4x + 5
2. Determine the operation: Addition
3. Perform the operation: (f + g)(x) = (3x^2 + 2x - 1) + (x^2 - 4x + 5)
4. Simplify the expression: (f + g)(x) = 3x^2 + x^2 + 2x - 4x - 1 + 5 (f + g)(x) = 4x^2 - 2x + 4
5. Determine the domain: Since both f(x) and g(x) are polynomials, their domains are all real numbers. Therefore, the domain of (f + g)(x) is also all real numbers.

Example 2: Subtracting Polynomial Functions

Using the same functions as above, let's find (f - g)(x).

1. Identify the functions: f(x) = 3x^2 + 2x - 1 and g(x) = x^2 - 4x + 5
2. Determine the operation: Subtraction
3. Perform the operation: (f - g)(x) = (3x^2 + 2x - 1) - (x^2 - 4x + 5)
4. Simplify the expression: (f - g)(x) = 3x^2 + 2x - 1 - x^2 + 4x - 5 (f - g)(x) = 3x^2 - x^2 + 2x + 4x - 1 - 5 (f - g)(x) = 2x^2 + 6x - 6
5. Determine the domain: Since both f(x) and g(x) are polynomials, their domains are all real numbers. Therefore, the domain of (f - g)(x) is also all real numbers.

Example 3: Adding Functions with Restrictions on the Domain

Suppose we have two functions:

• f(x) = √(x - 2)
• g(x) = √(5 - x)

We want to find (f + g)(x) and its domain.

1. Identify the functions: f(x) = √(x - 2) and g(x) = √(5 - x)

2. Determine the operation: Addition

3. Perform the operation: (f + g)(x) = √(x - 2) + √(5 - x)

4. Simplify the expression: The expression is already in its simplest form.

5. Determine the domain:

   • For f(x) = √(x - 2), the domain is x ≥ 2 (because we can't take the square root of a negative number).
   • For g(x) = √(5 - x), the domain is x ≤ 5.
   • The domain of (f + g)(x) is the intersection of these two domains: 2 ≤ x ≤ 5.

Example 4: Subtracting Functions with Rational Expressions

Let's consider two rational functions:

• f(x) = (x + 1) / (x - 2)
• g(x) = (x - 3) / (x + 1)

We want to find (f - g)(x) and its domain.

1. Identify the functions: f(x) = (x + 1) / (x - 2) and g(x) = (x - 3) / (x + 1)

2. Determine the operation: Subtraction

3. Perform the operation: (f - g)(x) = (x + 1) / (x - 2) - (x - 3) / (x + 1)

4. Simplify the expression:

   • Find a common denominator: (x - 2)(x + 1)
   • Rewrite the fractions with the common denominator:
     • f(x) = [(x + 1)(x + 1)] / [(x - 2)(x + 1)] = (x^2 + 2x + 1) / [(x - 2)(x + 1)]
     • g(x) = [(x - 3)(x - 2)] / [(x + 1)(x - 2)] = (x^2 - 5x + 6) / [(x + 1)(x - 2)]
   • Subtract the fractions: (f - g)(x) = [(x^2 + 2x + 1) - (x^2 - 5x + 6)] / [(x - 2)(x + 1)] (f - g)(x) = (x^2 + 2x + 1 - x^2 + 5x - 6) / [(x - 2)(x + 1)] (f - g)(x) = (7x - 5) / [(x - 2)(x + 1)]

5. Determine the domain:

   • For f(x), the domain is all real numbers except x = 2.
   • For g(x), the domain is all real numbers except x = -1.
   • Therefore, the domain of (f - g)(x) is all real numbers except x = 2 and x = -1.

Domain Considerations

The domain of the sum or difference of two functions is a critical aspect of these operations. The domain of the resulting function is the intersection of the domains of the original functions. This is because both f(x) and g(x) must be defined for the sum or difference to be defined.

Example 5: Domain with Different Types of Functions

Let's say we have:

• f(x) = log(x + 3)
• g(x) = √(4 - x)

We want to find the domain of (f + g)(x).

1. Domain of f(x) = log(x + 3):

   • The argument of a logarithm must be positive, so x + 3 > 0, which means x > -3.

2. Domain of g(x) = √(4 - x):

   • The expression inside the square root must be non-negative, so 4 - x ≥ 0, which means x ≤ 4.

3. Intersection of Domains:

   • The domain of (f + g)(x) is the intersection of x > -3 and x ≤ 4, which is -3 < x ≤ 4.

Properties of Adding and Subtracting Functions

Adding and subtracting functions share some properties with addition and subtraction of numbers.

• Commutative Property of Addition: (f + g)(x) = (g + f)(x). The order in which you add functions does not matter.
• Associative Property of Addition: ((f + g) + h)(x) = (f + (g + h))(x). When adding multiple functions, the grouping does not matter.
• Not Commutative for Subtraction: (f - g)(x) ≠ (g - f)(x) in general. The order matters for subtraction.
• Distributive Property (with Scalar Multiplication): c(f + g)(x) = c f(x) + c g(x), where c is a constant.

Graphical Interpretation

Adding and subtracting functions can also be visualized graphically. To find the value of (f + g)(x) at a specific point x, you add the y-values of f(x) and g(x) at that point. Similarly, to find the value of (f - g)(x) at a specific point x, you subtract the y-value of g(x) from the y-value of f(x) at that point.

Example 6: Graphical Addition

Suppose we have the graphs of f(x) and g(x). To graph (f + g)(x):

1. Choose points: Select several x-values.
2. Find y-values: For each x-value, find the corresponding y-values of f(x) and g(x) from their graphs.
3. Add y-values: Add the y-values to get the y-value of (f + g)(x) at that point.
4. Plot points: Plot the new points (x, (f + g)(x)) on a new graph.
5. Connect the points: Connect the points to create the graph of (f + g)(x).

The same process applies to subtraction, but you subtract the y-values instead.

Applications of Adding and Subtracting Functions

Adding and subtracting functions have numerous applications in various fields:

1. Economics: Combining cost and revenue functions to find profit functions.
2. Physics: Analyzing the net effect of forces acting on an object.
3. Engineering: Modeling the combined effect of multiple signals or systems.
4. Computer Science: Combining algorithms to create more complex programs.
5. Environmental Science: Modeling the combined effect of different environmental factors.

Example 7: Economic Application

Let C(x) be the cost function for producing x units of a product, and let R(x) be the revenue function for selling x units. The profit function, P(x), is the difference between the revenue and cost:

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

If R(x) = 100x - 0.1x^2 and C(x) = 50x + 1000, then:

P(x) = (100x - 0.1x^2) - (50x + 1000) P(x) = 100x - 0.1x^2 - 50x - 1000 P(x) = -0.1x^2 + 50x - 1000

This profit function can be used to determine the number of units that maximize profit.

Example 8: Physics Application

Suppose an object is subject to two forces, F1(t) and F2(t), where t is time. The net force, F_net(t), acting on the object is the sum of these forces:

F_net(t) = F1(t) + F2(t)

If F1(t) = 10t and F2(t) = -5t^2, then:

F_net(t) = 10t - 5t^2

This net force function can be used to analyze the object's motion.

Common Mistakes to Avoid

When adding and subtracting functions, it's important to avoid common mistakes:

1. Incorrectly Distributing the Negative Sign: When subtracting functions, remember to distribute the negative sign to all terms in the second function. For example, (f - g)(x) = f(x) - (ax + b) = f(x) - ax - b, not f(x) - ax + b.
2. Forgetting to Find the Intersection of Domains: The domain of the resulting function is the intersection of the domains of the original functions. Always determine the domain of each function and find their intersection.
3. Algebraic Errors: Be careful when simplifying expressions, especially with fractions and radicals.
4. Incorrectly Combining Like Terms: Double-check that you are combining like terms correctly.

Advanced Topics

Once you have a solid understanding of adding and subtracting functions, you can explore more advanced topics:

1. Composition of Functions: Combining functions by plugging one function into another.
2. Inverse Functions: Finding functions that "undo" each other.
3. Transformations of Functions: Shifting, stretching, and reflecting functions.
4. Calculus: Using derivatives and integrals to analyze the behavior of functions.

Practice Problems

To solidify your understanding, try these practice problems:

1. Let f(x) = 2x^2 - 3x + 1 and g(x) = -x^2 + 5x - 4. Find (f + g)(x) and (f - g)(x). Determine their domains.
2. Let f(x) = √(x + 1) and g(x) = √(9 - x). Find (f + g)(x) and its domain.
3. Let f(x) = (x + 2) / (x - 1) and g(x) = (x - 4) / (x + 2). Find (f - g)(x) and its domain.
4. If C(x) = 60x + 2000 and R(x) = 120x - 0.2x^2, find the profit function P(x) = R(x) - C(x).
5. Graph f(x) = x and g(x) = x^2. Then, graph (f + g)(x).

Conclusion

Adding and subtracting functions are essential operations in mathematics. They allow us to combine functions in meaningful ways, creating new functions with unique properties and behaviors. By understanding the definitions, steps, and properties involved, you can effectively manipulate functions and apply them to real-world problems. Mastering this concept is crucial for success in higher-level mathematics and various fields that rely on mathematical modeling. Remember to pay attention to the domain of the resulting functions and avoid common mistakes to ensure accuracy in your calculations. Keep practicing and exploring more advanced topics to deepen your understanding and expand your mathematical toolkit.