Evaluate F(x) = 10 - 2x At X = 7

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Evaluate f(x) = 10 - 2x at x = 7

Hey guys! Today, we're diving into a super basic but crucial concept in mathematics: evaluating functions. Specifically, we're going to take a look at the function f(x) = 10 - 2x and figure out what happens when we plug in x = 7. This is a foundational skill, and once you've nailed it, you'll find it pops up everywhere from algebra to calculus. So, let's break it down step by step and make sure everyone's on the same page. Trust me, it's way easier than it sounds!

Understanding Functions

Before we jump into the problem, let's quickly recap what a function actually is. Think of a function like a machine. You feed it an input (in our case, a value for x), and it spits out an output (the value of f(x)). The function f(x) = 10 - 2x is a simple rule that tells us exactly what to do with the input x. In this case, it says: "Take x, multiply it by -2, and then add 10." Simple enough, right? Understanding this basic idea is key to being able to evaluate any function, no matter how complex it looks. So, if you're ever feeling lost, just remember the "input-output machine" analogy, and you'll be golden!

Now, why is understanding functions so important? Well, functions are the building blocks of mathematical models. They allow us to describe relationships between different quantities and make predictions about how things will change. Whether you're calculating the trajectory of a rocket, predicting the stock market, or designing a bridge, you're using functions. Mastering the basics, like evaluating functions at specific points, is crucial for building a solid foundation in math and science. It's like learning the alphabet before you can write a novel – you gotta start somewhere!

And it's not just about abstract math. Functions are all around us in the real world. Think about the relationship between the amount of gas you put in your car and how far you can drive. That's a function! Or the relationship between the temperature outside and how much your heating bill will be. Another function! By understanding how functions work, you can start to see the mathematical patterns that govern the world around you. So, let's get back to our example and see how we can apply this knowledge to solve a specific problem.

Evaluating f(7)

Alright, let's get down to business. We're given the function f(x) = 10 - 2x, and we want to find f(7). This means we need to substitute x = 7 into the function and simplify. Here's how we do it:

  1. Replace x with 7:

    f(7) = 10 - 2(7)

  2. Perform the multiplication:

    f(7) = 10 - 14

  3. Subtract:

    f(7) = -4

And that's it! We've found that f(7) = -4. See, I told you it was easier than it looks! The key is to take it one step at a time and carefully follow the order of operations (PEMDAS/BODMAS). Don't try to do too much in your head, and always double-check your work. A little bit of attention to detail can save you from making silly mistakes.

So, what does this result actually mean? Well, it tells us that when the input to our function is 7, the output is -4. In other words, the point (7, -4) lies on the graph of the function f(x) = 10 - 2x. This is a fundamental concept in graphing functions, and it's something you'll use again and again in your math journey. By understanding how to evaluate functions at specific points, you can start to build a visual picture of what the function looks like and how it behaves.

Common Mistakes to Avoid

Even though evaluating functions is relatively straightforward, there are a few common mistakes that students often make. Here are a few things to watch out for:

  • Forgetting the order of operations: Always remember to perform multiplication before addition or subtraction. In our example, you need to multiply -2 by 7 before you subtract from 10.
  • Sign errors: Be careful with negative signs! It's easy to make a mistake when you're dealing with negative numbers, so double-check your work.
  • Incorrect substitution: Make sure you're substituting the correct value for x. It sounds obvious, but it's easy to get mixed up, especially when you're working with more complex functions.
  • Trying to do too much in your head: Write out each step clearly to avoid making mistakes. It might take a little longer, but it's worth it in the long run.

By being aware of these common pitfalls, you can avoid making them yourself and ensure that you're getting the correct answer every time. Remember, practice makes perfect! The more you practice evaluating functions, the more comfortable and confident you'll become.

Practice Problems

Now that we've worked through an example, let's try a few practice problems to solidify your understanding. Here are a couple of functions for you to evaluate:

  1. g(x) = 3x + 5. Find g(2).
  2. h(x) = x^2 - 4x + 1. Find h(0).
  3. k(x) = -5x - 7. Find k(-1).

Take your time, work through each problem step by step, and double-check your answers. If you get stuck, go back and review the example we worked through earlier. Remember, the key is to practice and build your confidence.

Real-World Applications

Okay, so we've learned how to evaluate functions, but why is this actually useful in the real world? Well, as I mentioned earlier, functions are used to model all sorts of relationships between different quantities. Here are a few examples:

  • Physics: The distance an object travels can be modeled as a function of time. By evaluating this function, you can predict how far the object will travel after a certain amount of time.
  • Economics: The demand for a product can be modeled as a function of its price. By evaluating this function, you can predict how many units of the product will be sold at a certain price.
  • Computer science: Algorithms are essentially functions that take inputs and produce outputs. By evaluating these functions, you can understand how the algorithm will behave for different inputs.

These are just a few examples, but the possibilities are endless. Functions are a fundamental tool in any field that involves mathematical modeling.

Conclusion

So, there you have it! We've successfully evaluated the function f(x) = 10 - 2x at x = 7 and found that f(7) = -4. We've also discussed why understanding functions is so important and how they're used in the real world. I hope this has been a helpful and informative guide. Keep practicing, and you'll be a function-evaluating pro in no time! Remember, math is all about building a solid foundation and taking it one step at a time. You got this!

Final Answer: The final answer is (A)