Evaluating Functions F(x) = -2x² - 3x + 4 And G(x) = 3/(2(x-1))

Introduction

Hey guys! In this article, we're going to dive into evaluating functions, specifically two functions: f(x) = -2x² - 3x + 4 and g(x) = 3/(2(x-1)). We'll tackle two questions: finding the value of f(-1) and determining the value of g(1/2). This is a fundamental concept in mathematics, and mastering it will set you up for success in more advanced topics. So, let's jump right in and break down these problems step by step. Understanding how to evaluate functions is crucial for various mathematical applications, from graphing and calculus to real-world problem-solving. When you substitute a value for x into a function, you are essentially asking, "What is the output of this function when the input is this specific value?" The process involves replacing every instance of x in the function's expression with the given value and then simplifying the resulting expression using the order of operations. This might involve performing arithmetic operations, squaring terms, or even simplifying fractions. By the end of this article, you will have a solid grasp of how to evaluate functions confidently and accurately.

Determining f(-1)

So, first up, let's figure out f(-1). Remember, f(x) = -2x² - 3x + 4. What we need to do here is substitute x with -1 in the function. This means wherever we see an x, we're going to replace it with (-1). Let's walk through it together. We start with the function f(x) = -2x² - 3x + 4. Now, substitute x with -1: f(-1) = -2(-1)² - 3(-1) + 4. Next, we need to follow the order of operations (PEMDAS/BODMAS). First, we handle the exponent: (-1)² = 1. So, the expression becomes: f(-1) = -2(1) - 3(-1) + 4. Now, let's do the multiplication: -2(1) = -2 and -3(-1) = 3. The expression now looks like this: f(-1) = -2 + 3 + 4. Finally, we perform the addition: -2 + 3 = 1, and then 1 + 4 = 5. Therefore, f(-1) = 5. See? It's not as scary as it looks! By carefully substituting and following the order of operations, we can easily find the value of the function at a specific point. Remember, the key is to take it one step at a time and double-check your work. When dealing with negative numbers and exponents, it's especially important to pay attention to the signs. A small mistake in the sign can lead to a completely different answer. So, always double-check your calculations and make sure you're following the correct order of operations. With practice, evaluating functions will become second nature, and you'll be able to tackle even more complex problems with confidence.

Determining g(1/2)

Alright, now let's move on to the second part: finding g(1/2). Our function here is g(x) = 3/(2(x-1)). Just like before, we're going to substitute x with 1/2. So, let's do it! We have g(x) = 3/(2(x-1)). Substituting x with 1/2, we get: g(1/2) = 3/(2(1/2 - 1)). First, let's simplify inside the parentheses: 1/2 - 1 = -1/2. So, the expression becomes: g(1/2) = 3/(2(-1/2)). Next, we perform the multiplication in the denominator: 2(-1/2) = -1. Now we have: g(1/2) = 3/(-1). Finally, we do the division: 3/(-1) = -3. Therefore, g(1/2) = -3. Excellent! We've successfully evaluated another function. Working with fractions might seem a bit tricky at first, but with practice, you'll become more comfortable with them. The key is to break down the problem into smaller steps and tackle each part one at a time. Remember to pay close attention to the order of operations and to simplify fractions whenever possible. When dealing with fractions inside fractions, it can be helpful to think of the division as multiplying by the reciprocal. For example, if you have a fraction in the denominator, you can multiply both the numerator and the denominator by the reciprocal of the denominator to simplify the expression. This technique can be particularly useful when evaluating more complex functions. Keep practicing, and you'll be a pro at working with fractions in no time!

Conclusion

And there you have it, guys! We've successfully evaluated both f(-1) and g(1/2). We found that f(-1) = 5 and g(1/2) = -3. By understanding the process of substitution and following the order of operations, you can confidently evaluate any function. Remember, practice makes perfect, so keep working at it! Evaluating functions is a fundamental skill in mathematics, and it's essential for success in higher-level courses. It's also a skill that has many real-world applications, from modeling physical phenomena to analyzing data. The ability to evaluate functions allows you to make predictions, understand relationships, and solve problems in a variety of contexts. So, don't underestimate the importance of this skill! As you continue your mathematical journey, you'll encounter increasingly complex functions and problems. However, the basic principles of evaluation remain the same. By mastering these principles, you'll be well-equipped to tackle any challenge that comes your way. Keep practicing, keep exploring, and most importantly, keep having fun with math! Remember, math is not just about numbers and equations; it's about developing critical thinking skills and problem-solving abilities that will serve you well in all aspects of life.