Evaluating H(x) -4x-7 At X 3 A Step-by-Step Guide

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In this article, we will delve into the process of evaluating the function h(x) = -4x - 7 when x = 3. This is a fundamental concept in algebra and calculus, and understanding it is crucial for solving more complex mathematical problems. We will break down the steps involved, provide clear explanations, and offer additional insights to ensure you grasp the concept thoroughly. This comprehensive guide will serve as a valuable resource for students, educators, and anyone interested in refreshing their understanding of function evaluation. The ability to accurately evaluate functions is a cornerstone of mathematical proficiency, enabling you to solve equations, model real-world phenomena, and make informed decisions based on mathematical relationships. So, let's embark on this journey together and unlock the power of function evaluation.

H2: Understanding Function Notation

Before we begin, let's clarify what the notation h(x) = -4x - 7 means. In mathematical terms, h(x) represents a function, which is essentially a rule that assigns a unique output value for every input value. The 'x' inside the parentheses is the input, also known as the independent variable, and h(x) represents the output, or the dependent variable. The equation -4x - 7 defines the rule or the operation that is applied to the input 'x' to produce the output h(x). This particular function is a linear function, characterized by its straight-line graph. The coefficient -4 represents the slope of the line, indicating the rate of change of the output with respect to the input, and the constant term -7 represents the y-intercept, the point where the line crosses the vertical axis. Understanding this basic structure of a function is crucial for evaluating it at any given point. When we say we want to evaluate h(x) when x = 3, we are essentially asking what the output value will be when we substitute 3 for 'x' in the function's equation. This process involves replacing every instance of 'x' in the equation with the value 3 and then performing the necessary arithmetic operations to calculate the result. This fundamental concept underlies many mathematical applications, from solving equations to modeling real-world phenomena.

H2: Step-by-Step Evaluation of h(3)

Now, let's dive into the actual process of evaluating h(x) when x = 3. This involves a straightforward substitution and simplification process. Here’s how we do it step-by-step:

  1. Substitute x with 3: In the function h(x) = -4x - 7, replace every instance of 'x' with the value 3. This gives us h(3) = -4(3) - 7.

  2. Perform the multiplication: Next, we perform the multiplication operation: -4 multiplied by 3 equals -12. So, our equation becomes h(3) = -12 - 7.

  3. Perform the subtraction: Finally, we perform the subtraction: -12 minus 7 equals -19. Therefore, h(3) = -19.

Thus, we have determined that the value of the function h(x) when x = 3 is -19. This means that when the input is 3, the output of the function is -19. This seemingly simple process is fundamental to understanding how functions work and how they can be used to model relationships between variables. By evaluating functions at specific points, we can gain insights into their behavior and use them to solve problems in various fields, including physics, engineering, economics, and computer science. The ability to confidently perform these steps is a key skill in mathematics and its applications.

H2: The Significance of Function Evaluation

Function evaluation is not just a mathematical exercise; it's a powerful tool with wide-ranging applications. Understanding how to evaluate functions allows us to predict outcomes, solve equations, and model real-world phenomena. For instance, in physics, a function might describe the trajectory of a projectile, and evaluating the function at a specific time would tell us the projectile's position at that moment. In economics, a function could represent the demand for a product, and evaluating it at a particular price would give us the quantity demanded. In computer science, functions are the building blocks of programs, and evaluating them is essential for executing the code. The process of function evaluation is also fundamental to solving equations. When we solve an equation, we are essentially finding the input values that make the function equal to a specific output value. This often involves evaluating the function at different points to see if we can find a solution. Furthermore, function evaluation plays a crucial role in calculus, particularly in the concepts of limits and derivatives. Limits describe the behavior of a function as the input approaches a certain value, and derivatives measure the rate of change of a function at a specific point. Both of these concepts rely heavily on the ability to evaluate functions accurately. In essence, function evaluation is a core skill in mathematics that underpins many other concepts and applications. Mastering this skill is essential for anyone pursuing further studies in mathematics, science, engineering, or any other field that relies on quantitative analysis.

H2: Common Mistakes to Avoid

While the process of evaluating a function is relatively straightforward, there are some common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure accurate results. One common mistake is incorrect substitution. When replacing 'x' with the given value, it's crucial to ensure that you substitute it correctly in every instance within the function's equation. For example, in the function h(x) = -4x - 7, if you forget to replace 'x' in the term -4x, you will get an incorrect result. Another frequent error is related to the order of operations. Remember to follow the PEMDAS/BODMAS rule (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction) when simplifying the expression after substitution. In our example, it's essential to perform the multiplication (-4 * 3) before the subtraction. A third common mistake involves sign errors. Be particularly careful when dealing with negative signs. For instance, in the equation h(3) = -4(3) - 7, the negative sign in front of the 4 is part of the coefficient and should not be ignored. Similarly, when subtracting negative numbers, remember that subtracting a negative is the same as adding a positive. Finally, it's always a good idea to double-check your work. After evaluating a function, take a moment to review each step to ensure that you haven't made any arithmetic errors. This simple practice can significantly improve your accuracy and prevent careless mistakes. By being mindful of these common pitfalls, you can enhance your understanding of function evaluation and achieve accurate results consistently.

H2: Practice Problems

To solidify your understanding of evaluating functions, let's work through a few practice problems. These examples will help you apply the concepts we've discussed and build your confidence in solving similar problems.

Problem 1: Evaluate the function f(x) = 2x² + 3x - 1 when x = -2.

Solution:

  1. Substitute x with -2: f(-2) = 2(-2)² + 3(-2) - 1
  2. Perform the exponentiation: f(-2) = 2(4) + 3(-2) - 1
  3. Perform the multiplications: f(-2) = 8 - 6 - 1
  4. Perform the additions and subtractions: f(-2) = 1

Therefore, f(-2) = 1.

Problem 2: Evaluate the function g(x) = (x + 5) / (x - 2) when x = 4.

Solution:

  1. Substitute x with 4: g(4) = (4 + 5) / (4 - 2)
  2. Simplify the numerator and denominator: g(4) = 9 / 2

Therefore, g(4) = 9/2 or 4.5.

Problem 3: Evaluate the function k(x) = √ (x + 6) when x = 3.

Solution:

  1. Substitute x with 3: k(3) = √ (3 + 6)
  2. Simplify inside the square root: k(3) = √ 9
  3. Evaluate the square root: k(3) = 3

Therefore, k(3) = 3.

These practice problems illustrate the fundamental steps involved in evaluating functions. Remember to substitute the given value correctly, follow the order of operations, and be mindful of signs. By working through a variety of examples, you can strengthen your understanding and improve your problem-solving skills. Consider creating your own practice problems with different functions and values to further challenge yourself and enhance your mastery of this essential mathematical concept.

H2: Conclusion

In conclusion, evaluating the function h(x) = -4x - 7 when x = 3 is a fundamental mathematical process that involves substituting the given value for the variable and simplifying the expression. We have demonstrated that h(3) = -19. This concept is essential for understanding and working with functions in various mathematical contexts and real-world applications. By mastering the steps involved in function evaluation, you will be well-equipped to tackle more complex problems and gain a deeper appreciation for the power of mathematical functions. Remember to practice regularly, pay attention to detail, and avoid common mistakes to ensure accurate results. With consistent effort and a solid understanding of the underlying principles, you can confidently evaluate functions and apply this skill to a wide range of situations. This article has provided a comprehensive guide to evaluating functions, and we hope it has been a valuable resource for your mathematical journey. Keep exploring, keep practicing, and keep unlocking the potential of mathematics!