Comparing Function Operations: Smallest X Coefficient

Hey guys! Let's dive into some cool math stuff. We're gonna explore functions and see how different operations on them affect the coefficients of the x term. Specifically, we have two functions: f(x) = -3x + 1 and g(x) = -2x - 3. Our mission? To figure out which operation—f + g, f - g, or f * g—results in the smallest coefficient attached to the x. It's like a little treasure hunt where x is the gold, and we're trying to find the operation that gives us the least amount of that gold. This is a great way to brush up on your algebra skills and understand how functions behave when we add, subtract, or multiply them. Ready to get started? Let's break it down step by step, making sure we understand each operation and its impact on our functions. We'll simplify and compare, and by the end, we'll have our answer. No sweat, this will be fun and enlightening! Understanding function operations is fundamental in mathematics and opens the door to more complex concepts, so let's make sure we've got a good grasp of the basics. We'll go through each operation systematically, calculate the resulting function, identify the x coefficient, and then compare the results. It's like a little detective work, tracking down those x coefficients. Get your pencils and calculators ready, because it's time to uncover the secrets of function manipulation! This is more than just finding the answer; it's about understanding the 'why' behind it. Let's see which operation gives us the smallest coefficient on that x term.

Analyzing the Sum of Functions: (f + g)(x)

Alright, first up, let's tackle the sum of our functions: (f + g)(x). This is pretty straightforward. All we need to do is add the two functions together. So, we have f(x) = -3x + 1 and g(x) = -2x - 3. Adding them is as simple as combining like terms. We take the x terms and add them together, and we take the constant terms and add them together. Let's do it! So, (f + g)(x) = (-3x + 1) + (-2x - 3). Now, let's simplify this. We combine the x terms: -3x + (-2x) = -5x. Then, we combine the constant terms: 1 + (-3) = -2. Therefore, (f + g)(x) = -5x - 2. The coefficient of the x term in this case is -5. This means that when we add the two functions, the x term becomes a more negative number. Remember, the smaller the number (in terms of coefficient), the smaller we are after the operation. Not too hard, right? It's all about combining like terms. We keep in mind that the goal is to identify the smallest coefficient, and we'll compare this -5 with the coefficients we get from the other operations. So, we have our first result! Let's move on to the next operation and see what happens when we subtract the functions. This is just the start, and it is a relatively basic process. Now, let's see what happens when we subtract the two functions.

Investigating the Difference of Functions: (f - g)(x)

Okay, now let's look at the difference of the functions, or (f - g)(x). This is where things can get a little tricky because we're subtracting, so we need to pay close attention to the signs. We'll subtract g(x) from f(x). Remember, f(x) = -3x + 1 and g(x) = -2x - 3. So, we're going to calculate (f - g)(x) = (-3x + 1) - (-2x - 3). Watch out for those minus signs! We need to distribute the negative sign to both terms in g(x). That means we're really adding the opposite of g(x). So, we have -(-2x) which becomes +2x, and -(-3) which becomes +3. Rewriting the equation, we get (f - g)(x) = -3x + 1 + 2x + 3. Now, let's combine the like terms. Combining the x terms, we have -3x + 2x = -x. Combining the constant terms, we have 1 + 3 = 4. Therefore, (f - g)(x) = -x + 4. The coefficient of the x term here is -1. So, in this operation, we get -1 for our coefficient. Notice how important it is to be careful with those negative signs during the subtraction. This step makes sure we don't make a simple mistake that throws off the whole calculation! We now have a second result to compare: -1. Now that we have the difference, we are ready to find the product of the functions, which will allow us to determine which operation gives us the smallest coefficient.

Examining the Product of Functions: (f * g)(x)

Alright, time to multiply our functions. This one's a bit more involved, but don't sweat it; we'll walk through it step by step. We're looking at (f * g)(x), which means we need to multiply f(x) = -3x + 1 by g(x) = -2x - 3. We'll use the distributive property (or the FOIL method, if you're familiar with that). This means we'll multiply each term in the first function by each term in the second function. Let's do it! First, multiply -3x by -2x, which gives us 6x². Next, multiply -3x by -3, which gives us +9x. Then, multiply 1 by -2x, which gives us -2x. Finally, multiply 1 by -3, which gives us -3. Putting it all together, we have 6x² + 9x - 2x - 3. Now, let's combine like terms. We only have one x² term, so that stays as 6x². Combining the x terms, we get 9x - 2x = 7x. The constant term is just -3. So, (f * g)(x) = 6x² + 7x - 3. The coefficient of the x term here is 7. This operation yields a positive 7 for the coefficient. Remember that we are seeking the smallest coefficient of the x-term. So, with the product of our functions, we've completed our calculations for all three operations: addition, subtraction, and multiplication. It's time to put on our detective hats and compare the results.

Comparing the Coefficients and Determining the Smallest Value

Time to put it all together! We've calculated the x coefficients for each of the three operations: addition, subtraction, and multiplication. Let's recap and compare:

• For (f + g)(x), the x coefficient was -5.
• For (f - g)(x), the x coefficient was -1.
• For (f * g)(x), the x coefficient was 7.

Now, remember that we're looking for the smallest coefficient. When we say 'smallest,' we're considering the value on the number line. Negative numbers are smaller than positive numbers, and the further to the left you go on the number line, the smaller the number. Comparing -5, -1, and 7, the smallest number is -5. Therefore, the operation that results in the smallest x coefficient is (f + g)(x), the sum of the two functions. So there you have it! We've successfully navigated through the different operations, calculated the resulting functions, and identified the one with the smallest x coefficient. Great job, everyone! Now, that you've seen how it works, you should practice on your own with a few different functions. This is a fundamental concept that will help you later on. Keep practicing, and you'll become a pro in no time!

Final Answer

The operation that results in the smallest coefficient on the x term is f + g.