Finding F(x-3) Given F(x) = X² - 4x + 3

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Hey guys! Today, we're diving into a super common and important concept in mathematics: function composition. Specifically, we're going to tackle a problem where we need to find F(x - 3) given that F(x) = x² - 4x + 3. This might seem a little daunting at first, but trust me, we'll break it down step-by-step so it's crystal clear. Think of it like this: we're not just plugging in a number for x; we're plugging in an entire expression! Ready to get started?

What is Function Composition?

Before we jump into the problem, let's quickly recap what function composition actually means. Imagine a function like a machine: you feed it an input, and it spits out an output. Now, function composition is like having two of these machines connected in a chain. The output of the first machine becomes the input of the second machine. In mathematical terms, if we have two functions, say f(x) and g(x), then the composition of f with g, written as f(g(x)), means we first evaluate g(x), and then we take that result and plug it into f(x). It’s like a mathematical relay race!

The key thing to remember is the order: we work from the inside out. In our case, we're dealing with F(x - 3). This means we're taking the expression (x - 3) and substituting it everywhere we see x in the original function F(x). This is a fundamental concept in algebra and calculus, so getting a solid grasp on it now will pay dividends later on. Think of it like building a strong foundation for a house; the stronger your foundation, the taller and more complex the house (or in this case, the math!) you can build. So, let’s make sure our foundation is rock solid!

Step-by-Step Solution for F(x - 3)

Okay, let’s get down to business and solve this problem! We're given that F(x) = x² - 4x + 3, and we want to find F(x - 3). Here's how we'll do it:

Step 1: Identify the Substitution

The first and most crucial step is to recognize what we're substituting. In this case, we're replacing every instance of x in the function F(x) with the expression (x - 3). It's like we're swapping out one thing for another. This might seem simple, but it's where a lot of mistakes can happen if we're not careful. So, let’s highlight this: we're replacing x with (x - 3). Keep this in mind as we move forward.

Step 2: Substitute (x - 3) into F(x)

Now comes the substitution part. We'll take the original function, F(x) = x² - 4x + 3, and replace each x with (x - 3). This gives us:

F(x - 3) = (x - 3)² - 4(x - 3) + 3

Notice how we've carefully placed the (x - 3) in parentheses. This is super important! The parentheses ensure that we're applying the operations correctly, especially when dealing with exponents and multiplication. Without parentheses, we might end up with the wrong answer. Think of parentheses like the walls of a building; they define the space and ensure everything is in its proper place.

Step 3: Expand and Simplify

Now that we've made the substitution, the next step is to expand and simplify the expression. This involves a bit of algebraic manipulation, but nothing we can't handle! First, let's expand the squared term, (x - 3)². Remember, this means (x - 3) * (x - 3). We can use the FOIL method (First, Outer, Inner, Last) or the distributive property to expand this:

(x - 3)² = (x - 3)(x - 3) = x² - 3x - 3x + 9 = x² - 6x + 9

Next, let's distribute the -4 in the term -4(x - 3):

-4(x - 3) = -4x + 12

Now, we can rewrite F(x - 3) with these expanded terms:

F(x - 3) = x² - 6x + 9 - 4x + 12 + 3

Finally, let's combine like terms to simplify the expression. We have terms, x terms, and constant terms. Combining them gives us:

F(x - 3) = x² - 6x - 4x + 9 + 12 + 3 = x² - 10x + 24

And there you have it! We've successfully found F(x - 3). It might seem like a lot of steps, but each one is manageable on its own. The key is to be organized and careful with your algebra. Think of it like baking a cake; each ingredient needs to be measured and added in the right order for the final product to be delicious!

Common Mistakes to Avoid

Before we wrap up, let's talk about some common mistakes people make when dealing with function composition. Being aware of these pitfalls can help you avoid them in the future.

Forgetting Parentheses

As we mentioned earlier, parentheses are crucial! When substituting an expression into a function, always use parentheses to ensure you're applying the operations in the correct order. Forgetting parentheses can lead to incorrect expansion and simplification.

Incorrectly Expanding Squared Terms

A common mistake is to think that (x - 3)² is equal to x² - 3² (which is x² - 9). This is wrong! Remember that (x - 3)² means (x - 3)(x - 3), and we need to use FOIL or the distributive property to expand it correctly.

Sign Errors

Be extra careful with signs, especially when distributing negative numbers. A simple sign error can throw off your entire solution. Double-check your work, and pay close attention to those negative signs!

Combining Unlike Terms

Make sure you're only combining like terms. You can't add terms to x terms or constant terms. It's like trying to add apples and oranges; they're different things!

Practice Makes Perfect

The best way to master function composition is to practice, practice, practice! Try working through similar problems with different functions and expressions. The more you practice, the more comfortable and confident you'll become. Remember, math is like a muscle; the more you exercise it, the stronger it gets!

Conclusion

So, guys, we've successfully navigated the world of function composition and found F(x - 3) given F(x) = x² - 4x + 3. We've covered the basics of function composition, worked through a step-by-step solution, and highlighted some common mistakes to avoid. Remember, the key is to take it one step at a time, be organized, and pay attention to detail. With a little practice, you'll be a function composition pro in no time!

If you have any questions or want to try some more examples, feel free to ask. Keep practicing, and keep exploring the fascinating world of mathematics! You got this!