Evaluating F(x+4) For F(x) = X^2 - X: A Step-by-Step Guide

by Alex Johnson 59 views

In this article, we'll break down how to evaluate the composite function f(x+4) when given the function f(x) = x^2 - x. This is a common type of problem in algebra and calculus, and understanding how to solve it is crucial for mastering function operations. We'll go through the process step-by-step, ensuring you grasp the underlying concepts. So, let's dive in!

Understanding Function Composition

Before we jump into the specifics of f(x+4), let's briefly discuss function composition. Function composition is essentially plugging one function into another. In our case, we're not plugging in a numerical value, but rather an expression, x+4, into the function f(x). This means that wherever we see x in the original function, we will replace it with the expression x+4. Think of it as a substitution process where the entire x+4 expression takes the place of x in the function's formula. It's important to remember that this substitution has to be done consistently throughout the entire function. Understanding this concept will make the evaluation process much smoother. Function composition is a fundamental concept, and mastering it will help you tackle more complex problems in mathematics. Let's proceed with a straightforward approach to make sure every detail is clear, and you feel confident in handling these evaluations. So, keep reading to learn the step-by-step method that simplifies everything.

Step-by-Step Evaluation of f(x+4)

Here’s how we evaluate f(x+4) when f(x) = x^2 - x:

Step 1: Replace x with (x+4)

The first step is to replace every instance of x in the function f(x) with the expression (x+4). This means wherever you see x, you'll substitute it with (x+4). So, f(x) = x^2 - x becomes f(x+4) = (x+4)^2 - (x+4). This step is crucial because it sets up the entire evaluation process. Make sure you're substituting the entire expression (x+4), including the parentheses. Failing to do so can lead to errors in the next steps. Pay close attention to where x appears in the original function, and carefully replace it with (x+4). This simple substitution is the foundation for simplifying the expression and finding the final result. Once you've correctly substituted, you're ready to move on to the next step, which involves expanding and simplifying the expression.

Step 2: Expand (x+4)^2

Next, we need to expand the term (x+4)^2. Recall that (x+4)^2 is the same as (x+4)(x+4). To expand this, we use the FOIL method (First, Outer, Inner, Last) or the distributive property. (x+4)(x+4) = xx + x4 + 4x + 44 = x^2 + 4x + 4x + 16 = x^2 + 8x + 16. Therefore, (x+4)^2 expands to x^2 + 8x + 16. It's important to remember this expansion, as it’s a common algebraic identity. Knowing this will not only speed up your calculations but also help you avoid mistakes. Expanding the squared term correctly is essential for simplifying the expression and getting to the final answer. Once you've expanded (x+4)^2, you can substitute this expansion back into the expression and proceed to the next step. This careful expansion ensures that you're working with the correct terms, paving the way for accurate simplification.

Step 3: Distribute the Negative Sign

Now, let's address the second part of the expression: -(x+4). To simplify this, we distribute the negative sign to both terms inside the parentheses. This means that -(x+4) becomes -x - 4. Remember, when distributing a negative sign, each term inside the parentheses changes its sign. So, +x becomes -x, and +4 becomes -4. This is a straightforward but crucial step, as overlooking the negative sign can lead to incorrect results. Be careful to apply the negative sign to every term inside the parentheses. Once you've correctly distributed the negative sign, the expression becomes simpler and easier to manage. This step is vital for ensuring that all terms are accounted for and properly signed before combining like terms in the next step. Accurate distribution of the negative sign is a key element in arriving at the correct final answer.

Step 4: Combine Like Terms

Now that we've expanded (x+4)^2 and distributed the negative sign, our expression looks like this: f(x+4) = x^2 + 8x + 16 - x - 4. The next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In this expression, we have the following like terms: 8x and -x, and 16 and -4. Combining 8x and -x, we get 7x. Combining 16 and -4, we get 12. So, the simplified expression becomes f(x+4) = x^2 + 7x + 12. This step is essential for reducing the expression to its simplest form. Be careful to only combine terms that are alike. Once you've combined all like terms, you'll have the final, simplified expression for f(x+4). This step ensures that the expression is as concise as possible, making it easier to work with in subsequent calculations or applications.

Step 5: Final Result

After combining all like terms, we arrive at the final result: f(x+4) = x^2 + 7x + 12. This is the simplified expression for f(x+4) when f(x) = x^2 - x. This result represents the composite function, showing how the original function changes when x is replaced by x+4. To summarize, we started by replacing x with (x+4) in the original function, expanded the squared term, distributed the negative sign, and then combined like terms to arrive at the final simplified expression. This step-by-step process ensures accuracy and clarity in evaluating composite functions. The final result, x^2 + 7x + 12, is a quadratic expression that can be further analyzed or used in other mathematical contexts. Understanding how to arrive at this result is crucial for mastering function operations and algebra.

Example

Let’s see an example with f(x) = x^2 - x.

Find f(x+4).

  1. Replace x with (x+4): f(x+4) = (x+4)^2 - (x+4)
  2. Expand: f(x+4) = x^2 + 8x + 16 - x - 4
  3. Combine like terms: f(x+4) = x^2 + 7x + 12

Therefore, f(x+4) = x^2 + 7x + 12.

Common Mistakes to Avoid

When evaluating composite functions, it's easy to make mistakes. Here are a few common pitfalls to watch out for:

  • Incorrectly Expanding (x+4)^2: A common mistake is to incorrectly expand (x+4)^2 as x^2 + 4^2 instead of x^2 + 8x + 16. Remember to use the FOIL method or the distributive property to correctly expand the square of a binomial.
  • Forgetting to Distribute the Negative Sign: When dealing with expressions like -(x+4), it's crucial to distribute the negative sign to both terms inside the parentheses. Failing to do so will result in an incorrect expression.
  • Combining Unlike Terms: Only combine terms that have the same variable raised to the same power. For example, you can combine 8x and -x, but you cannot combine x^2 and 7x.
  • Not Substituting Correctly: Ensure that you replace every instance of x in the original function with the expression (x+4). Missing a substitution can lead to an incorrect result.

By being aware of these common mistakes and taking the time to double-check your work, you can avoid errors and confidently evaluate composite functions.

Practice Problems

To solidify your understanding, here are a couple of practice problems:

  1. If f(x) = 2x^2 + 3x - 1, find f(x-2).
  2. If g(x) = x^3 - 4x, find g(x+1).

Work through these problems, applying the steps we've discussed, and check your answers to ensure you've mastered the process. Practice makes perfect when it comes to evaluating composite functions. So, take the time to work through these problems carefully and reinforce your understanding.

Conclusion

Evaluating composite functions like f(x+4) might seem daunting at first, but by following a step-by-step approach, you can simplify the process and arrive at the correct answer. Remember to replace x with the given expression, expand and simplify, and combine like terms. With practice and attention to detail, you'll become proficient in evaluating composite functions. By understanding these concepts and applying them consistently, you'll build a strong foundation in algebra and calculus. This skill is not only useful for academic purposes but also has applications in various fields, including engineering, computer science, and economics. So, keep practicing and honing your skills to excel in mathematics!

For further learning about functions, you can visit Khan Academy's Functions and Equations Section.