Evaluate H(-67) For H(x) = -49x - 125
Introduction
In this article, we will delve into the process of evaluating a function at a specific point. Specifically, we will focus on finding the value of h(-67) for the function h(x) = -49x - 125. This involves substituting x with -67 in the given function and performing the necessary arithmetic operations to arrive at the final answer. Understanding how to evaluate functions is a fundamental skill in mathematics, particularly in algebra and calculus. It allows us to determine the output of a function for a given input, which is crucial for analyzing the behavior of functions and their applications in various real-world scenarios. We will break down the steps involved in this evaluation, ensuring a clear and concise understanding of the process. This article aims to provide a comprehensive guide on how to evaluate a function at a specific point, using the example of h(x) = -49x - 125 and x = -67. By the end of this article, you will be able to confidently evaluate similar functions for any given input value. The ability to evaluate functions is not only essential for solving mathematical problems but also for understanding concepts in physics, engineering, and other scientific disciplines where mathematical models are used to describe and predict phenomena.
Understanding Function Evaluation
Before we dive into the specific calculation, let's clarify the concept of function evaluation. A function, in mathematical terms, is a relationship between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. We often represent a function using the notation f(x), where x is the input variable, and f(x) represents the output of the function for that input. Evaluating a function at a specific point means substituting the input variable x with a given value and simplifying the expression to find the corresponding output. For instance, if we have a function f(x) = 2x + 3, evaluating it at x = 2 would involve replacing x with 2 in the expression, resulting in f(2) = 2(2) + 3 = 7. This process of substitution and simplification is the core of function evaluation. It's crucial to follow the order of operations (PEMDAS/BODMAS) to ensure accurate results. Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). A thorough understanding of function evaluation is essential for various mathematical concepts, including graphing functions, solving equations, and understanding transformations. It also lays the groundwork for more advanced topics like calculus, where the concept of a limit relies heavily on function evaluation. In essence, function evaluation is a fundamental building block in mathematics, enabling us to analyze and interpret mathematical relationships effectively.
The Function h(x) = -49x - 125
In this particular problem, we are given the function h(x) = -49x - 125. This is a linear function, which means that its graph will be a straight line. The function is defined for all real numbers, meaning that we can input any real number for x, and the function will produce a corresponding output. The expression -49x represents the product of -49 and the input value x, while -125 is a constant term. To evaluate this function at a specific point, we will substitute the value of x into the expression and simplify. For example, if we wanted to find h(0), we would substitute x with 0, resulting in h(0) = -49(0) - 125 = -125. The coefficient -49 in the function represents the slope of the line, indicating the rate of change of the function's output with respect to its input. The constant term -125 represents the y-intercept, which is the point where the line crosses the y-axis. Understanding the components of a linear function, such as the slope and y-intercept, can provide valuable insights into its behavior and characteristics. Linear functions are widely used in various fields, including economics, physics, and engineering, to model relationships between variables that exhibit a constant rate of change. The function h(x) = -49x - 125 is a simple yet powerful example of a linear function, illustrating the fundamental principles of mathematical modeling.
Substituting x = -67 into h(x)
Now, let's proceed with the task at hand: finding the value of h(-67). This involves substituting x with -67 in the function h(x) = -49x - 125. When substituting, it's crucial to pay close attention to signs and parentheses to avoid errors. Replacing x with -67, we get: h(-67) = -49(-67) - 125. Notice that we've placed -67 in parentheses to clearly indicate that it's being multiplied by -49. This is particularly important when dealing with negative numbers to ensure correct calculations. The next step is to perform the multiplication and subtraction operations according to the order of operations. Remember, multiplying two negative numbers results in a positive number. Therefore, -49(-67) will be a positive value. This substitution process is a fundamental aspect of function evaluation and is applicable to various types of functions, not just linear functions. Whether dealing with quadratic, exponential, or trigonometric functions, the principle remains the same: replace the input variable with the given value and simplify the expression. Accurate substitution is the foundation for successful function evaluation, and careful attention to detail is essential to avoid mistakes. In the following section, we will perform the arithmetic calculations to arrive at the final value of h(-67).
Performing the Arithmetic
Having substituted x = -67 into the function, we now have the expression: h(-67) = -49(-67) - 125. The first step is to perform the multiplication: -49 multiplied by -67. As mentioned earlier, the product of two negative numbers is a positive number. So, we have: (-49) * (-67) = 3283. Now, our expression becomes: h(-67) = 3283 - 125. The next step is to perform the subtraction: 3283 minus 125. Subtracting 125 from 3283, we get: 3283 - 125 = 3158. Therefore, the value of h(-67) is 3158. It is essential to perform these arithmetic operations accurately, following the order of operations. Mistakes in multiplication or subtraction can lead to an incorrect final answer. Using a calculator can help ensure accuracy, especially when dealing with larger numbers. The result we obtained, 3158, represents the output of the function h(x) = -49x - 125 when the input is x = -67. This value corresponds to a specific point on the graph of the linear function. In the next section, we will summarize our findings and present the final answer.
Final Answer
After substituting x = -67 into the function h(x) = -49x - 125 and performing the necessary arithmetic operations, we have arrived at the final answer. The calculations are as follows:
h(-67) = -49(-67) - 125 h(-67) = 3283 - 125 h(-67) = 3158
Therefore, the value of h(-67) is 3,158. This means that when the input to the function h(x) is -67, the output is 3,158. This result represents a specific point on the graph of the linear function h(x) = -49x - 125. Function evaluation is a crucial skill in mathematics, allowing us to determine the output of a function for a given input. In this case, we have successfully evaluated the function h(x) at x = -67, demonstrating the process of substitution and simplification. The final answer, 3,158, is the solution to the problem and represents the value of the function at the specified input. This exercise highlights the importance of careful arithmetic and attention to detail when working with mathematical functions. With a solid understanding of function evaluation, you can confidently tackle more complex mathematical problems and applications.
Conclusion
In conclusion, we have successfully found the value of h(-67) for the function h(x) = -49x - 125. By substituting x with -67 and performing the arithmetic operations, we determined that h(-67) = 3,158. This process involved understanding the concept of function evaluation, substituting the input variable with the given value, and simplifying the expression according to the order of operations. Function evaluation is a fundamental skill in mathematics, and mastering it is crucial for various applications in algebra, calculus, and other scientific disciplines. This exercise has demonstrated a clear and concise approach to evaluating a linear function at a specific point. The steps involved include careful substitution, accurate arithmetic calculations, and attention to detail. The final answer, 3,158, represents the output of the function for the given input and provides a specific point on the function's graph. By understanding and practicing function evaluation, you can confidently solve similar problems and gain a deeper understanding of mathematical relationships. The ability to evaluate functions is not only essential for academic success but also for real-world applications where mathematical models are used to describe and predict phenomena.
