Evaluating F(-1) - G(-1) For Given Functions

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Hey guys! Let's dive into a math problem where we need to evaluate the expression f(-1) - g(-1) given two functions: f(x) = -6x + 17 and g(x) = x^2 + 14. This might sound a bit daunting at first, but don't worry, we'll break it down step by step. We'll start by understanding what these functions mean and how to substitute values into them. Then, we'll calculate f(-1) and g(-1) individually before finally subtracting the results to get our answer. So, grab your calculators (or your trusty mental math skills) and let's get started!

Understanding the Functions

Before we jump into calculations, let's make sure we understand what these functions, f(x) and g(x), actually represent. In simple terms, a function is like a little machine that takes an input (in this case, x) and produces an output based on a specific rule. For f(x) = -6x + 17, the rule is: multiply the input x by -6, and then add 17. For g(x) = x^2 + 14, the rule is: square the input x, and then add 14.

Think of it like this: if we put a number into the f machine, it gets multiplied by -6 and then has 17 added to it. If we put the same number into the g machine, it gets squared and then has 14 added to it. Our goal here is to figure out what happens when we put -1 into both machines, and then subtract the output of the g machine from the output of the f machine. It’s all about following the rules and keeping track of our numbers!

Remember, the x in f(x) and g(x) is just a placeholder. It tells us where to put the input value in the expression. So, when we see f(-1), it means we replace every x in the expression for f(x) with -1. This is a crucial concept, and once you've got it down, the rest is just arithmetic. So, let’s move on to the next step where we’ll actually calculate these values.

Calculating f(-1)

Alright, let's start by figuring out the value of f(-1). Remember, f(x) = -6x + 17. To find f(-1), we simply substitute -1 for x in the expression. So, we get: f(-1) = -6(-1) + 17. Now, let's break this down step by step.

First, we need to multiply -6 by -1. Remember the rule: a negative times a negative equals a positive. So, -6 * (-1) = 6. Now our expression looks like this: f(-1) = 6 + 17. Next, we simply add 6 and 17. 6 + 17 = 23. Therefore, f(-1) = 23.

See? It's not so bad when you take it one step at a time. We replaced x with -1, did the multiplication, and then the addition. We've successfully calculated the output of the f machine when the input is -1. This is a big step, and we're well on our way to solving the whole problem. Now that we've found f(-1), let's move on to calculating g(-1). We'll follow a similar process, substituting -1 into the expression for g(x) and working through the arithmetic.

Calculating g(-1)

Okay, now let's tackle g(-1). We know that g(x) = x^2 + 14. Just like with f(x), we're going to substitute -1 for x in this expression. So, we have g(-1) = (-1)^2 + 14. Remember that the exponent means we need to square -1, which means multiplying -1 by itself.

So, let's calculate (-1)^2 first. (-1) * (-1) = 1. A negative number multiplied by a negative number is always positive. Now our expression becomes: g(-1) = 1 + 14. This is a simple addition problem. 1 + 14 = 15. Therefore, g(-1) = 15.

Great job! We've now figured out the output of the g machine when the input is -1. We squared -1 to get 1, and then added 14. We're making excellent progress! We've calculated both f(-1) and g(-1) individually. Now, the final step is to subtract g(-1) from f(-1), which will give us the answer to the original problem. Let's head to the next section and wrap this up.

Finding f(-1) - g(-1)

We're in the home stretch now! We've already calculated that f(-1) = 23 and g(-1) = 15. The final step is to find f(-1) - g(-1). This simply means subtracting the value of g(-1) from the value of f(-1). So, we have: f(-1) - g(-1) = 23 - 15.

This is a straightforward subtraction problem. 23 - 15 = 8. Therefore, f(-1) - g(-1) = 8. And that's it! We've successfully evaluated the expression. We started by understanding the functions, then we calculated f(-1) and g(-1) separately, and finally, we subtracted the results.

We’ve conquered this math problem together! Remember, the key is to break down complex problems into smaller, manageable steps. By substituting the value, following the order of operations, and keeping track of our calculations, we were able to find the solution. Let's recap the steps we took and highlight the key concepts we used.

Recap and Key Concepts

Let's quickly recap what we did to solve this problem. First, we understood the functions f(x) = -6x + 17 and g(x) = x^2 + 14. We recognized that these functions are like machines that take an input (x) and produce an output based on a specific rule.

Next, we calculated f(-1) by substituting -1 for x in the expression for f(x). We got f(-1) = -6(-1) + 17 = 6 + 17 = 23. Then, we calculated g(-1) by substituting -1 for x in the expression for g(x). We got g(-1) = (-1)^2 + 14 = 1 + 14 = 15.

Finally, we found f(-1) - g(-1) by subtracting g(-1) from f(-1). We got f(-1) - g(-1) = 23 - 15 = 8. So, the final answer is 8.

The key concepts we used here are:

  • Function notation: Understanding that f(x) represents a function of x, and substituting a value for x means replacing every instance of x in the expression with that value.
  • Order of operations: Following the correct order of operations (PEMDAS/BODMAS) when evaluating expressions (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction).
  • Integer arithmetic: Performing operations with positive and negative numbers, including multiplication and squaring.

By mastering these concepts, you'll be well-equipped to tackle similar problems in the future. Keep practicing, and remember to break down complex problems into smaller, more manageable steps. You've got this! Math can be fun and rewarding when you approach it with confidence and a step-by-step strategy.