Solving Functions: Find F(x) + G(x) - H(x)

by Alex Johnson 43 views

Let's dive into the world of functions and explore how to combine them! In this article, we'll break down the process of finding the result of f(x) + g(x) - h(x) when given three different functions. This is a fundamental concept in mathematics, and understanding it will help you tackle more complex problems.

Understanding the Functions

Before we jump into the calculation, let's make sure we understand the functions we're working with. We have three functions:

  • f(x) = 3x + 5
  • g(x) = 4x² - 2
  • h(x) = x² - 3x + 1

Each of these functions takes an input x and performs a specific set of operations to produce an output. Our goal is to combine these functions through addition and subtraction.

Step-by-Step Calculation

Now, let's calculate f(x) + g(x) - h(x) step by step. This involves adding the expressions for f(x) and g(x), and then subtracting the expression for h(x).

1. Write Out the Expression

First, let's write out the full expression:

f(x) + g(x) - h(x) = (3x + 5) + (4x² - 2) - (x² - 3x + 1)

This clearly shows how we'll be combining the functions.

2. Combine f(x) and g(x)

Next, we'll add f(x) and g(x):

(3x + 5) + (4x² - 2) = 4x² + 3x + 5 - 2

Simplifying this, we get:

4x² + 3x + 3

3. Subtract h(x)

Now, we subtract h(x) from the result we just obtained:

(4x² + 3x + 3) - (x² - 3x + 1)

Distribute the negative sign to each term in h(x):

4x² + 3x + 3 - x² + 3x - 1

4. Combine Like Terms

Finally, we combine like terms:

  • Combine the x² terms: 4x² - x² = 3x²
  • Combine the x terms: 3x + 3x = 6x
  • Combine the constant terms: 3 - 1 = 2

Putting it all together, we get:

3x² + 6x + 2

The Result

Therefore, f(x) + g(x) - h(x) = 3x² + 6x + 2.

So, the correct answer is B. 3x² + 6x + 2.

Why is this Important?

Understanding how to combine functions is crucial in many areas of mathematics and its applications. Here are a few reasons why:

  • Modeling Real-World Situations: Functions are used to model various real-world phenomena. Combining functions allows us to create more complex and accurate models. For example, you might use one function to model the cost of materials and another to model labor costs. Adding these functions together would give you a total cost function.
  • Calculus: Calculus deals with rates of change and accumulation. Understanding how functions interact is essential for concepts like derivatives and integrals.
  • Computer Science: In programming, functions are the building blocks of programs. Combining functions is a fundamental part of writing efficient and complex code.
  • Physics and Engineering: Many physical laws and engineering principles are expressed using functions. Combining functions allows engineers and physicists to analyze and design complex systems.

Common Mistakes to Avoid

When working with functions, there are a few common mistakes that students often make. Being aware of these pitfalls can help you avoid them.

  • Incorrectly Distributing the Negative Sign: When subtracting a function, it's crucial to distribute the negative sign to every term inside the parentheses. For example, -(x² - 3x + 1) becomes -x² + 3x - 1, not -x² - 3x + 1.
  • Combining Unlike Terms: You can only combine terms that have the same variable and exponent. For example, you can combine 3x² and -x², but you can't combine 3x² and 6x.
  • Forgetting to Simplify: After performing the operations, always simplify your expression by combining like terms. This ensures you have the most concise and accurate answer.
  • Misunderstanding Function Notation: Make sure you understand what the function notation means. For example, f(x) means "the value of the function f at x," not "f times x."

Practice Makes Perfect

The best way to master combining functions is to practice! Try working through various examples with different types of functions. Here are a few practice problems you can try:

  1. Given f(x) = 2x - 1, g(x) = x² + 3, and h(x) = 4x - 2, find f(x) + g(x) - h(x).
  2. Given f(x) = x³ + 2x, g(x) = -x² + 5, and h(x) = 3x² - x, find f(x) - g(x) + h(x).
  3. Given f(x) = 5x + 4, g(x) = 2x² - 3x, and h(x) = x² + x - 1, find f(x) + g(x) + h(x).

By working through these problems, you'll build your skills and confidence in combining functions.

Real-World Applications and Deeper Understanding

To truly appreciate the power of combining functions, let's explore some real-world applications and delve deeper into the underlying concepts.

Modeling Costs and Revenue

In business, functions are often used to model costs, revenue, and profit. For example:

  • C(x) might represent the cost of producing x units of a product.
  • R(x) might represent the revenue generated from selling x units.

The profit, P(x), can then be found by subtracting the cost function from the revenue function:

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

This simple equation demonstrates how combining functions can provide valuable insights into business operations.

Analyzing Motion in Physics

In physics, functions are used to describe motion. For instance:

  • s(t) might represent the position of an object at time t.
  • v(t) might represent the velocity of the object at time t.

If we know the initial position and velocity of an object, we can use functions to predict its future position. Combining functions can help us analyze more complex scenarios, such as the motion of a projectile under the influence of gravity and air resistance.

Creating Complex Computer Graphics

In computer graphics, functions are used to define shapes, colors, and movements. Combining functions allows artists and programmers to create intricate and realistic visual effects. For example, blending different color functions can produce gradients, and combining transformation functions can create animations.

Conclusion

Combining functions is a fundamental skill in mathematics with applications across various fields. By understanding the basic principles and practicing regularly, you can master this concept and unlock its potential. Remember to pay attention to details like distributing negative signs and combining like terms. With a solid understanding of combining functions, you'll be well-equipped to tackle more advanced mathematical challenges.

For further exploration and practice, you might find resources on websites like Khan Academy's Function Operations extremely helpful. Happy function combining!