Evaluating Functions A Step-by-Step Guide With F(x) = -2|5-x|-13

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In the realm of mathematics, functions serve as fundamental building blocks, describing relationships between inputs and outputs. Mastering the art of evaluating functions is crucial for students venturing into algebra, calculus, and beyond. This article delves into the process of evaluating functions, using the example f(x)=βˆ’2∣5βˆ’xβˆ£βˆ’13f(x) = -2|5-x|-13 to illustrate the key steps involved. We will dissect the function, substitute the given value, and perform the necessary arithmetic operations to arrive at the final answer. By the end of this exploration, you will have a solid understanding of how to confidently evaluate functions, regardless of their complexity.

Understanding Function Notation and Evaluation

Before we embark on evaluating the specific function, it's essential to grasp the fundamental concepts of function notation and evaluation. A function, denoted as f(x)f(x), establishes a relationship between an input value (represented by xx) and an output value (represented by f(x)f(x)). The expression f(x)f(x) is read as "f of x" and signifies the value of the function ff when the input is xx. To evaluate a function at a particular value, we substitute that value for xx in the function's expression and simplify the resulting expression.

For instance, consider the function f(x)=x2+3xβˆ’2f(x) = x^2 + 3x - 2. To evaluate f(2)f(2), we replace every instance of xx with 2: f(2)=(2)2+3(2)βˆ’2=4+6βˆ’2=8f(2) = (2)^2 + 3(2) - 2 = 4 + 6 - 2 = 8. This signifies that when the input is 2, the output of the function ff is 8. Function evaluation is a cornerstone of mathematical analysis, providing insights into the behavior of functions and their applications in various fields.

Dissecting the Function f(x)=βˆ’2∣5βˆ’xβˆ£βˆ’13f(x) = -2|5-x|-13

Now, let's turn our attention to the function at hand: f(x)=βˆ’2∣5βˆ’xβˆ£βˆ’13f(x) = -2|5-x|-13. This function comprises several components, each playing a distinct role in determining the output value. The function involves an absolute value expression, ∣5βˆ’x∣|5-x|, which represents the distance of (5βˆ’x)(5-x) from zero. The absolute value ensures that the result is always non-negative. The expression (5βˆ’x)(5-x) is then multiplied by -2, and finally, 13 is subtracted from the result. Understanding the order of operations within the function is crucial for accurate evaluation. We must first evaluate the expression inside the absolute value, then take the absolute value, followed by multiplication, and lastly, subtraction. This step-by-step approach ensures that we arrive at the correct output value for any given input.

Evaluating f(6)f(6) Step-by-Step

Our goal is to evaluate f(6)f(6), which means finding the output value of the function when the input is 6. To achieve this, we will substitute 6 for xx in the function's expression and simplify systematically.

  1. Substitution: Replace every instance of xx in the function with 6:

    f(6)=βˆ’2∣5βˆ’6βˆ£βˆ’13f(6) = -2|5-6|-13

  2. Evaluate inside the absolute value: Simplify the expression inside the absolute value bars:

    f(6)=βˆ’2βˆ£βˆ’1βˆ£βˆ’13f(6) = -2|-1|-13

  3. Evaluate the absolute value: The absolute value of -1 is 1:

    f(6)=βˆ’2(1)βˆ’13f(6) = -2(1)-13

  4. Multiplication: Multiply -2 by 1:

    f(6)=βˆ’2βˆ’13f(6) = -2-13

  5. Subtraction: Subtract 13 from -2:

    f(6)=βˆ’15f(6) = -15

Therefore, f(6)=βˆ’15f(6) = -15. When the input is 6, the output of the function ff is -15. This step-by-step process illustrates how we can systematically evaluate functions by substituting the given value and simplifying the resulting expression, adhering to the order of operations.

Common Mistakes to Avoid

Evaluating functions might seem straightforward, but certain common mistakes can lead to incorrect results. One frequent error is neglecting the order of operations. Remember to prioritize operations within parentheses or absolute value symbols, followed by multiplication and division, and finally, addition and subtraction. Another pitfall is incorrectly evaluating the absolute value. The absolute value of a number is its distance from zero, which is always non-negative. For instance, βˆ£βˆ’3∣=3|-3| = 3, not -3. A third mistake is overlooking the negative sign in front of the absolute value. In the function f(x)=βˆ’2∣5βˆ’xβˆ£βˆ’13f(x) = -2|5-x|-13, the -2 multiplies the absolute value, so it's crucial to include the negative sign in the calculation. By being mindful of these common errors and practicing diligently, you can enhance your accuracy in evaluating functions.

Practice Problems and Solutions

To solidify your understanding of function evaluation, let's tackle a few more practice problems. Consider the function g(x)=3x2βˆ’4x+1g(x) = 3x^2 - 4x + 1. Evaluate g(βˆ’2)g(-2).

  1. Substitution: g(βˆ’2)=3(βˆ’2)2βˆ’4(βˆ’2)+1g(-2) = 3(-2)^2 - 4(-2) + 1

  2. Exponents: g(βˆ’2)=3(4)+8+1g(-2) = 3(4) + 8 + 1

  3. Multiplication: g(βˆ’2)=12+8+1g(-2) = 12 + 8 + 1

  4. Addition: g(βˆ’2)=21g(-2) = 21

Therefore, g(βˆ’2)=21g(-2) = 21.

Now, let's evaluate h(x) = rac{2x + 5}{x - 3} at x=4x = 4.

  1. Substitution: h(4) = rac{2(4) + 5}{4 - 3}

  2. Multiplication: h(4) = rac{8 + 5}{1}

  3. Addition: h(4) = rac{13}{1}

  4. Division: h(4)=13h(4) = 13

Hence, h(4)=13h(4) = 13. These examples demonstrate the consistent application of the substitution and simplification process in function evaluation. Consistent practice with diverse functions is key to mastering this skill.

Real-World Applications of Function Evaluation

Function evaluation is not merely an abstract mathematical exercise; it has profound real-world applications across various disciplines. In physics, functions are used to model the motion of objects, and evaluating these functions at specific times allows us to determine the object's position or velocity at that instant. In economics, functions can represent cost, revenue, or profit, and evaluating them helps businesses make informed decisions about pricing and production levels. In computer science, functions are the building blocks of programs, and evaluating them is essential for executing code and obtaining results. For instance, a function might calculate the area of a circle given its radius, and evaluating this function with different radii provides the corresponding areas. The ability to evaluate functions bridges the gap between mathematical models and real-world phenomena, making it an indispensable skill in numerous fields.

Conclusion: Mastering Function Evaluation

In conclusion, evaluating functions is a fundamental skill in mathematics with wide-ranging applications. By understanding function notation, dissecting the function's expression, and following a systematic step-by-step approach, you can confidently evaluate functions of varying complexity. Remember to pay close attention to the order of operations and avoid common mistakes. Consistent practice and exposure to diverse function types will solidify your understanding and enhance your proficiency. Whether you're a student delving into mathematical concepts or a professional applying mathematical models, mastering function evaluation will empower you to solve problems and make informed decisions. So, embrace the power of functions, and let them be your tools for unraveling the intricacies of the world around you.

By understanding function notation, dissecting functions into their components, and practicing a systematic step-by-step approach, we can confidently evaluate functions of varying complexity. This article provided a comprehensive guide to evaluating functions, using the example f(x)=βˆ’2∣5βˆ’xβˆ£βˆ’13f(x) = -2|5-x|-13. We explored the underlying concepts, walked through the evaluation process step-by-step, highlighted common mistakes to avoid, and provided practice problems to solidify understanding. With consistent practice and attention to detail, you can master function evaluation and unlock its power in various mathematical and real-world applications. Remember, the journey of learning mathematics is a continuous one, and each step forward brings you closer to a deeper understanding of the world around us.