Evaluating Composite Functions A Step-by-Step Solution
In mathematics, functions are fundamental building blocks that describe relationships between inputs and outputs. Often, we encounter scenarios where we need to combine multiple functions to create more complex operations. This involves evaluating composite functions, where the output of one function becomes the input of another. This article provides a comprehensive guide to evaluating composite functions, with a focus on a specific example involving three functions: f(x) = 3x + 1, g(x) = -2x^2, and h(x) = (1/3)√(x) - 1. We will break down the process step by step, ensuring a clear understanding of the concepts involved. Mastering composite functions is crucial for various mathematical disciplines, including calculus, algebra, and analysis. Composite functions allow us to model complex relationships by combining simpler functions, providing a powerful tool for problem-solving. In this article, we will not only focus on the mechanical steps of evaluating the expression but also on the underlying principles of function composition. This understanding will empower you to tackle a wide range of problems involving composite functions. The ability to manipulate and evaluate composite functions is essential for further studies in mathematics and related fields. From understanding the behavior of complex systems to solving intricate equations, composite functions play a vital role. This article aims to equip you with the necessary skills and knowledge to confidently work with composite functions, paving the way for advanced mathematical concepts. Let's delve into the world of composite functions and unravel the process of evaluating them.
Understanding the Functions
Before diving into the evaluation, let's clearly define the functions we'll be working with. We have three functions:
- f(x) = 3x + 1: This is a linear function, where the output is obtained by multiplying the input (x) by 3 and adding 1.
- g(x) = -2x²: This is a quadratic function, where the output is obtained by squaring the input (x), multiplying by -2.
- h(x) = (1/3)√(x) - 1: This function involves a square root. The output is obtained by taking the square root of the input (x), multiplying by 1/3, and subtracting 1.
Each of these functions performs a specific operation on its input. To evaluate the expression (1/16)f(-1) - h(81) + g(2), we need to evaluate each function at the given input values and then combine the results according to the expression. It's crucial to understand the nature of each function – whether it's linear, quadratic, or involves a square root – as this will guide the order of operations and the calculations involved. For instance, in the function g(x), squaring the input before multiplying by -2 is essential for obtaining the correct output. Similarly, in the function h(x), taking the square root before multiplying by 1/3 and subtracting 1 is crucial. The order of operations is paramount when dealing with functions, and understanding the specific operations each function performs will ensure accurate evaluation. These functions, while seemingly simple on their own, can be combined to create more complex expressions, highlighting the power of functional notation in mathematics. By mastering the evaluation of individual functions, we can confidently tackle composite expressions involving multiple functions, paving the way for more advanced mathematical concepts. Let's now proceed to evaluate each function at its respective input value.
Evaluating f(-1)
To evaluate f(-1), we substitute -1 for x in the function f(x) = 3x + 1:
f(-1) = 3(-1) + 1
f(-1) = -3 + 1
f(-1) = -2
Therefore, the value of f(-1) is -2. This step involves a straightforward substitution and arithmetic calculation. We replace the variable 'x' in the function's expression with the given input value, which in this case is -1. The substitution is a fundamental step in evaluating functions, as it allows us to determine the output corresponding to a specific input. Following the substitution, we perform the arithmetic operations according to the order of operations (PEMDAS/BODMAS). In this case, we first multiply 3 by -1, resulting in -3, and then add 1 to obtain the final result of -2. This simple example illustrates the core concept of function evaluation, where we map an input value to an output value based on the function's definition. The result, f(-1) = -2, signifies that when the input to the function f(x) is -1, the output is -2. This mapping is the essence of a function, and understanding how to perform this evaluation is crucial for working with more complex functions and expressions. Now that we have evaluated f(-1), we can use this result in the overall expression we aim to evaluate. Let's move on to the next function, h(x), and evaluate it at the given input value.
Evaluating h(81)
Next, we evaluate h(81) using the function h(x) = (1/3)√(x) - 1. We substitute 81 for x:
h(81) = (1/3)√(81) - 1
Since the square root of 81 is 9, we have:
h(81) = (1/3)(9) - 1
h(81) = 3 - 1
h(81) = 2
So, h(81) equals 2. This evaluation involves an additional step compared to the previous one, as we need to calculate the square root of the input before proceeding with the other operations. The square root function, denoted by √(x), returns the non-negative value that, when multiplied by itself, equals x. In this case, the square root of 81 is 9 because 9 * 9 = 81. Once we have the square root, we multiply it by 1/3, which gives us (1/3) * 9 = 3. Finally, we subtract 1 from the result, leading to the final output of 2. This step-by-step process highlights the importance of following the order of operations when evaluating functions that involve multiple operations. The presence of the square root function adds a layer of complexity, but by carefully applying the order of operations, we can accurately determine the output. The result, h(81) = 2, signifies that when the input to the function h(x) is 81, the output is 2. This evaluation demonstrates the application of a function involving a square root, a common type of function encountered in mathematics. Now that we have evaluated h(81), we can proceed to evaluate the third function, g(x), at its given input value.
Evaluating g(2)
Now, let's evaluate g(2) using the function g(x) = -2x²:
g(2) = -2(2²)
First, we square 2, which gives us 4:
g(2) = -2(4)
Then, we multiply -2 by 4:
g(2) = -8
Therefore, g(2) is -8. This evaluation involves squaring the input value before multiplying by the coefficient. The squaring operation, denoted by x², means multiplying the input value by itself. In this case, 2² equals 2 * 2, which is 4. It's crucial to perform the squaring operation before multiplying by -2, as the order of operations dictates that exponents are evaluated before multiplication. After squaring the input, we multiply the result by -2, which gives us -2 * 4 = -8. The negative sign plays a significant role in this evaluation, as it changes the sign of the final output. The result, g(2) = -8, signifies that when the input to the function g(x) is 2, the output is -8. This evaluation demonstrates the application of a quadratic function, a common type of function characterized by its squared term. Now that we have evaluated all three individual functions at their respective input values, we have all the necessary components to evaluate the entire expression. Let's move on to combining these results to find the final answer.
Combining the Results
Now that we have the values of f(-1), h(81), and g(2), we can substitute them into the expression (1/16)f(-1) - h(81) + g(2):
(1/16)f(-1) - h(81) + g(2) = (1/16)(-2) - 2 + (-8)
First, we multiply (1/16) by -2:
(1/16)(-2) = -2/16 = -1/8
Now, substitute this back into the expression:
-1/8 - 2 + (-8)
To combine these terms, we need a common denominator. We can rewrite 2 and 8 as fractions with a denominator of 8:
-1/8 - 16/8 - 64/8
Now, we can add the numerators:
(-1 - 16 - 64) / 8 = -81/8
Therefore, the value of the expression (1/16)f(-1) - h(81) + g(2) is -81/8. This step involves combining the results obtained from evaluating the individual functions, following the order of operations specified in the expression. We first multiply the fraction (1/16) by the value of f(-1), which is -2. This gives us -2/16, which can be simplified to -1/8. Next, we substitute the values of h(81) and g(2), which are 2 and -8, respectively. Now, we have the expression -1/8 - 2 + (-8). To combine these terms, we need to express them with a common denominator, which is 8. We rewrite 2 as 16/8 and 8 as 64/8. This allows us to add the numerators while keeping the denominator the same. Adding the numerators, we get -1 - 16 - 64 = -81. Therefore, the final result is -81/8. This result represents the value of the entire expression when the functions are evaluated at their respective input values and combined according to the given formula. The process of combining the results highlights the importance of accurate arithmetic and attention to detail. A single error in any of the previous steps could lead to an incorrect final answer. This comprehensive example showcases the step-by-step approach to evaluating composite functions and expressions.
Final Answer
The value of the expression (1/16)f(-1) - h(81) + g(2) is -81/8. This final answer represents the culmination of all the previous steps, where we evaluated each function at its given input value and then combined the results according to the expression's formula. The process involved a combination of arithmetic operations, including multiplication, subtraction, and addition of fractions. The result, -81/8, is a fraction, which is a common outcome when dealing with composite functions and expressions. It's important to express the final answer in its simplest form, if possible, but in this case, -81/8 is already in its simplest form as the numerator and denominator have no common factors other than 1. This comprehensive example demonstrates the entire process of evaluating a complex expression involving multiple functions. From understanding the individual functions to combining their results, each step is crucial for obtaining the correct final answer. The ability to confidently evaluate such expressions is a valuable skill in mathematics and related fields. By mastering the techniques presented in this article, you can tackle a wide range of problems involving composite functions and expressions.
In conclusion, this article has provided a detailed walkthrough of evaluating a composite function expression. We began by defining the individual functions, f(x), g(x), and h(x), and then proceeded to evaluate each function at its respective input value. We then combined the results, following the order of operations, to arrive at the final answer of -81/8. This process highlights the importance of understanding function notation, substitution, and arithmetic operations. The ability to evaluate composite functions is a fundamental skill in mathematics, with applications in various fields such as calculus, algebra, and computer science. By mastering this skill, you can unlock a deeper understanding of mathematical relationships and problem-solving techniques. The step-by-step approach presented in this article provides a clear and concise method for tackling similar problems. Remember to carefully evaluate each function individually, paying attention to the order of operations, and then combine the results according to the given expression. With practice and a solid understanding of the underlying concepts, you can confidently evaluate complex composite function expressions. This article serves as a valuable resource for students and anyone interested in enhancing their mathematical skills. By applying the techniques and principles discussed, you can successfully navigate the world of composite functions and expressions, opening doors to more advanced mathematical concepts and applications. The journey through the realm of functions and their compositions is a rewarding one, leading to a deeper appreciation of the elegance and power of mathematics.
