Evaluating And Solving The Function F(x) = 8 - 3x

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Hey guys! Today, we're diving deep into the world of functions, specifically the function f(x) = 8 - 3x. We're going to tackle two key aspects: evaluating the function at a specific point and solving for x when the function equals a certain value. Buckle up, because we're about to make math fun and accessible!

(a) Evaluating f(-2): Plugging in the Value

So, the first task at hand is to evaluate f(-2). What does this even mean? Well, in simple terms, it means we need to figure out what the function f(x) spits out when we feed it the input x = -2. Think of a function like a machine: you put something in (the input), and the machine does its thing and gives you something out (the output).

In this case, our machine is defined by the equation f(x) = 8 - 3x. To evaluate f(-2), we're going to replace every instance of x in the equation with the value -2. Let's break it down step-by-step:

  1. Start with the function: f(x) = 8 - 3x
  2. Substitute x with -2: f(-2) = 8 - 3(-2)

Now, we need to simplify the expression on the right-hand side. Remember the order of operations (PEMDAS/BODMAS)? We need to handle the multiplication before the subtraction.

  1. Multiply -3 by -2: f(-2) = 8 + 6 (Remember, a negative times a negative is a positive!)
  2. Add 8 and 6: f(-2) = 14

And there you have it! We've successfully evaluated f(-2) and found that it equals 14. This means that when we input -2 into our function machine, it outputs 14. Pretty cool, huh?

To further solidify this concept, let's think about why this process works. The function f(x) = 8 - 3x is a rule, a recipe, if you will. It tells us exactly what to do with any input x. We take the input, multiply it by -3, and then add 8. Evaluating the function at a specific point is simply following this rule for that particular input. This ability to evaluate functions is fundamental in mathematics and has applications in various fields, from physics and engineering to economics and computer science. Understanding how functions behave is crucial for modeling real-world phenomena and making predictions. So, mastering this skill is definitely worth your time and effort!

(b) Solving f(x) = -1: Finding the Input

Now, let's switch gears a bit. Instead of knowing the input and finding the output, we're going to do the reverse: we know the output and need to find the input. This is what it means to solve f(x) = -1. In this case, we're asking: what value of x do we need to put into our function machine so that it spits out -1?

Again, let's break it down step-by-step:

  1. Start with the function and the equation: f(x) = 8 - 3x and f(x) = -1
  2. Set the two expressions for f(x) equal to each other: 8 - 3x = -1 (Since both expressions equal f(x), they must equal each other!)

Now we have an algebraic equation that we need to solve for x. Our goal is to isolate x on one side of the equation. We'll do this by performing the same operations on both sides of the equation, maintaining the balance.

  1. Subtract 8 from both sides: 8 - 3x - 8 = -1 - 8 which simplifies to -3x = -9

  2. Divide both sides by -3: -3x / -3 = -9 / -3 which simplifies to x = 3

Fantastic! We've found the solution: x = 3. This means that when we input 3 into the function f(x) = 8 - 3x, the output is -1. We can double-check this by plugging 3 back into the original function:

f(3) = 8 - 3(3) = 8 - 9 = -1

Yep, it works! Solving equations like this is a cornerstone of algebra and is essential for countless applications. Whether you're determining the break-even point for a business, calculating the trajectory of a projectile, or optimizing a computer algorithm, solving equations is a skill you'll use time and time again. The ability to manipulate equations and isolate variables is a powerful tool for problem-solving in mathematics and beyond. So, practice makes perfect! Keep working on these types of problems, and you'll become a master equation solver in no time.

The Bigger Picture: Why Functions Matter

Okay, so we've tackled the specific problem of evaluating f(-2) and solving f(x) = -1 for the function f(x) = 8 - 3x. But let's zoom out for a second and think about the bigger picture: why do functions matter in the first place?

Functions, guys, are the workhorses of mathematics. They're the way we describe relationships between quantities. They allow us to model real-world phenomena, make predictions, and solve problems. Think about it: anything that has an input and an output can be represented by a function.

For example:

  • The temperature outside at a given time: The time is the input, and the temperature is the output.
  • The distance a car travels based on how long it's been driving: The driving time is the input, and the distance is the output.
  • The amount of money you earn based on the number of hours you work: The hours worked is the input, and the money earned is the output.

Functions come in all shapes and sizes. They can be linear (like the function we've been working with today), quadratic, exponential, trigonometric, and many more. Each type of function has its own unique properties and is useful for modeling different kinds of relationships.

Understanding the different types of functions and how they behave is key to unlocking the power of mathematics. When you can recognize a functional relationship, you can start to analyze it, make predictions, and solve problems related to it. This is why functions are so important in fields like science, engineering, economics, and computer science.

Mastering Functions: Practice and Persistence

So, where do we go from here? Well, the key to mastering functions, like any mathematical concept, is practice. The more you work with functions, the more comfortable you'll become with them. Try working through different examples, evaluating functions at various points, and solving equations involving functions. Don't be afraid to make mistakes – that's how we learn!

There are tons of resources available to help you practice: textbooks, online tutorials, practice problems, and even math games. Find what works best for you and stick with it. Consistency is key when learning math. Even just spending a few minutes each day working on functions can make a huge difference over time.

And remember, if you get stuck, don't hesitate to ask for help. Talk to your teacher, your classmates, or find a tutor. There are plenty of people who are happy to help you on your mathematical journey. Math can be challenging, but it's also incredibly rewarding. The more you understand it, the more you'll appreciate its power and beauty. Keep practicing, keep exploring, and keep having fun!

In conclusion, we've successfully evaluated f(-2) and solved f(x) = -1 for the function f(x) = 8 - 3x. More importantly, we've discussed the broader significance of functions in mathematics and their role in modeling real-world relationships. Remember, understanding functions is a fundamental skill that will serve you well in various fields. So, keep practicing, keep exploring, and embrace the power of functions! You got this!