Solving For X In F(x) = -x + 1 With Y Values -5, 0, And 5
Hey guys! Let's dive into a fun math problem today. We're going to figure out the value of x in the function f(x) = -x + 1 when y (which is the same as f(x)) takes on the values -5, 0, and 5. It might sound a bit complicated, but trust me, it's super straightforward once we break it down. We'll go through each value of y step by step, so you can see exactly how to solve these kinds of problems. Get ready to put on your math hats, and let's get started!
Understanding the Function f(x) = -x + 1
Before we jump into solving for x, let’s make sure we all understand what the function f(x) = -x + 1 really means. In simple terms, this function is a rule. It tells us exactly what to do with any input we give it, which we call x. The rule here is pretty straightforward: take the input x, multiply it by -1 (which changes its sign), and then add 1. The result we get after doing these operations is the output, which we call f(x). Think of it like a little machine: you feed it a number (x), it does some calculations, and then it spits out another number (f(x)).
Now, f(x) is just another way of saying y. So, when we talk about f(x) = -5, what we're really saying is y = -5. This is super important because it helps us connect the function notation to something we might be more familiar with, like the good old y = mx + b form of a linear equation. In our case, f(x) = -x + 1 is a linear equation with a slope of -1 and a y-intercept of 1. This means that for every increase of 1 in x, y decreases by 1. The y-intercept tells us that the line crosses the y-axis at the point (0, 1).
Understanding this function is like having a map before going on a journey. We know the terrain, the rules of the game, and what to expect. So, when we start solving for x, we're not just blindly plugging in numbers; we're using our understanding of the function to guide us. We know what the function does, and now we need to figure out how to reverse the process. Instead of finding y when we know x, we're going to find x when we know y. This is a bit like working backward, but it's totally doable, and we're going to master it together!
Solving for x When f(x) = -5
Alright, let's get our hands dirty and solve for x when f(x) = -5. Remember, f(x) is just another way of saying y, so we're really solving the equation -5 = -x + 1. Our mission here is to isolate x on one side of the equation. This means we want to get x all by itself, so we can see exactly what its value is. To do this, we need to undo the operations that are being done to x, but we need to do it in the reverse order.
The first thing we notice is that we're adding 1 to -x. To undo this addition, we need to subtract 1 from both sides of the equation. This is a golden rule in algebra: whatever you do to one side, you must do to the other. If we subtract 1 from both sides, our equation becomes:
-5 - 1 = -x + 1 - 1
Which simplifies to:
-6 = -x
We're getting closer! Now, we have -x on the right side, but we want plain old x. The - in front of the x means it's being multiplied by -1. To undo this multiplication, we need to divide both sides of the equation by -1. Remember the golden rule? We're applying it again! So, we divide both sides by -1:
-6 / -1 = -x / -1
This simplifies to:
6 = x
So, we've done it! We've found that when f(x) = -5, the value of x is 6. That wasn't so bad, was it? We just used our algebra skills to carefully undo the operations and isolate x. Now, let's keep the momentum going and tackle the next case where f(x) = 0.
Solving for x When f(x) = 0
Okay, guys, let's jump right into the next part of our adventure: solving for x when f(x) = 0. This time, we're tackling the equation 0 = -x + 1. Just like before, our main goal is to isolate x, which means getting it all by itself on one side of the equation. We'll use the same step-by-step approach we used earlier, carefully undoing the operations to reveal the value of x.
Looking at our equation, 0 = -x + 1, we see that 1 is being added to -x. To get rid of this addition, we need to do the opposite operation, which is subtraction. So, we'll subtract 1 from both sides of the equation. Remember that golden rule: what we do to one side, we have to do to the other. This ensures that our equation stays balanced and true. So, let's subtract 1 from both sides:
0 - 1 = -x + 1 - 1
This simplifies to:
-1 = -x
We're making good progress! Now we have -1 = -x. But we don't want -x; we want x. The negative sign in front of the x is like multiplying x by -1. To undo this, we need to do the opposite operation, which is division. So, we'll divide both sides of the equation by -1. Again, we're using the golden rule to keep everything balanced:
-1 / -1 = -x / -1
This simplifies to:
1 = x
Boom! We've solved it again. When f(x) = 0, the value of x is 1. See how we carefully followed the steps, undoing each operation one at a time? That's the key to solving these kinds of equations. Now, we have one more value of f(x) to tackle, and that's when f(x) = 5. Let's keep this momentum going and knock it out of the park!
Solving for x When f(x) = 5
Alright, team, we're on the home stretch! Let's solve for x one last time, this time when f(x) = 5. So, we're working with the equation 5 = -x + 1. By now, we're practically pros at this. We know the drill: our mission is to isolate x, and we'll do it by carefully undoing the operations, step by step. Remember our golden rule: what we do to one side, we gotta do to the other.
Looking at the equation 5 = -x + 1, we see that 1 is being added to -x. To undo this, we're going to subtract 1 from both sides. Let's do it:
5 - 1 = -x + 1 - 1
This simplifies to:
4 = -x
We're almost there! We've got 4 = -x, but we need to find x, not -x. The negative sign in front of the x is like multiplying x by -1. To undo this multiplication, we'll divide both sides by -1. Let's keep that golden rule in mind and apply it:
4 / -1 = -x / -1
This simplifies to:
-4 = x
Yes! We've nailed it. When f(x) = 5, the value of x is -4. We've successfully solved for x for all three given values of f(x). Give yourselves a pat on the back – you've earned it! We started by understanding the function, and then we systematically worked through each case, using our algebra skills to isolate x. Now, let's take a step back and summarize what we've learned.
Summary of Solutions
Okay, let's take a moment to recap what we've accomplished today, guys! We successfully solved for x in the function f(x) = -x + 1 for three different values of f(x) (which is the same as y). This kind of problem-solving is super important in math because it helps us understand how functions work and how to manipulate equations. Plus, it's a skill that comes in handy in all sorts of real-world situations.
Here’s a quick rundown of our solutions:
- When f(x) = -5, we found that x = 6
- When f(x) = 0, we found that x = 1
- When f(x) = 5, we found that x = -4
To find these solutions, we used a simple but powerful strategy: isolating x. This involved undoing the operations in the equation one by one, making sure to always do the same thing to both sides to keep the equation balanced. We subtracted, divided, and generally showed those equations who's boss! Remember that golden rule? It's your best friend when you're solving equations.
But more than just getting the right answers, we learned about the process of solving. We understood why we were doing each step, and that’s the kind of knowledge that sticks with you. You can apply this same strategy to solve all sorts of equations, no matter how intimidating they might look at first. So, keep practicing, keep asking questions, and keep exploring the wonderful world of math. You've got this!
Practice Problems
Now that we've worked through these examples together, it's your turn to shine! Practice makes perfect, as they say, so let's solidify your understanding with a few more problems. These are similar to what we just did, so you already have the tools you need to solve them. Grab a pencil and paper, and let's put your skills to the test!
Here are a few practice problems for you to try:
- Solve for x in f(x) = -x + 1 when f(x) = -2
- Solve for x in f(x) = -x + 1 when f(x) = 2
- Solve for x in f(x) = -x + 1 when f(x) = 10
Remember, the key is to isolate x by undoing the operations in the correct order. Start by adding or subtracting, and then multiply or divide. Don't forget that golden rule: whatever you do to one side, you must do to the other. And most importantly, don't be afraid to make mistakes! Mistakes are how we learn and grow. So, give these problems your best shot, and then check your answers.
Solving these practice problems is like building a muscle. Each time you work through a problem, you're strengthening your understanding and your problem-solving skills. So, take your time, be patient with yourself, and enjoy the process. And if you get stuck, don't hesitate to look back at our examples or ask for help. We're all in this together!
Conclusion
Great job, everyone! We've reached the end of our math adventure for today. We started with a function, f(x) = -x + 1, and we learned how to solve for x when given different values of f(x) (or y). We tackled equations, used our algebra skills, and even made a few mistakes along the way – which is totally okay! Remember, math isn't about being perfect; it's about learning, growing, and challenging ourselves.
We covered some key concepts, like understanding function notation, isolating variables, and the golden rule of equation solving. These are fundamental skills that will help you in all sorts of math problems, from simple algebra to more advanced topics. And the best part is, you now have a solid foundation to build on. You've seen how to approach these problems, and you've practiced the steps. You're well on your way to becoming equation-solving superstars!
But the journey doesn't end here. Keep exploring, keep practicing, and keep asking questions. Math is a fascinating world full of patterns, puzzles, and possibilities. And with each problem you solve, you're unlocking a little piece of that world. So, thank you for joining me on this adventure, and I can't wait to see what you'll conquer next! Keep up the great work, and remember, math is awesome!
