Finding Input X For H(x) = 5 - 2x Given Output 6

In the realm of mathematics, functions play a pivotal role in describing relationships between variables. Understanding how to work with functions, including finding input values for specific outputs, is a fundamental skill. This article will delve into the process of determining the input value (x) for a given function h(x) when the output is known. We will use the example function h(x) = 5 - 2x and the specific output value of 6 to illustrate the steps involved. This comprehensive guide aims to provide a clear and concise explanation, making it accessible to learners of all levels. Whether you're a student grappling with function concepts or simply seeking to refresh your mathematical skills, this article will equip you with the knowledge and confidence to tackle similar problems.

Understanding Functions

To effectively solve for the input of a function, it's crucial to first grasp the fundamental concept of what a function is. At its core, a function is a mathematical rule that assigns a unique output value for each input value. Think of it as a machine: you feed it an input, and it processes it according to a specific rule, delivering a corresponding output. In mathematical notation, we often represent a function as f(x), where x represents the input, and f(x) represents the output. The rule that connects x and f(x) can be expressed in various forms, such as an equation, a graph, or a table. Understanding this foundational concept is key to navigating the intricacies of function manipulation and problem-solving. For example, in our case, h(x) = 5 - 2x defines a linear function. The input x is multiplied by -2, and then the result is added to 5 to produce the output h(x). Visualizing this relationship as a machine helps to understand how changing the input x will affect the output h(x). This intuitive understanding forms the basis for solving problems where we need to find the input for a specific output.

Key Components of a Function

Delving deeper into the anatomy of a function, we can identify several key components that are essential for understanding its behavior and solving related problems. The input is the independent variable, often denoted as x, which we feed into the function. The output, also known as the dependent variable, is the value that the function produces based on the input. In the function notation f(x), f represents the name of the function, and the entire expression f(x) represents the output value. The rule of the function is the mathematical operation or set of operations that are applied to the input to generate the output. This rule can be expressed as an equation, such as h(x) = 5 - 2x, or in other forms like graphs or tables. The domain of a function is the set of all possible input values for which the function is defined. In other words, it's the set of all x values that can be plugged into the function without resulting in an undefined output (like division by zero or the square root of a negative number). The range of a function is the set of all possible output values that the function can produce. It's the set of all f(x) values that result from plugging in all the possible input values from the domain. Understanding these components – input, output, rule, domain, and range – provides a comprehensive framework for analyzing and working with functions. In the context of our problem, we are given the function rule (h(x) = 5 - 2x) and a specific output value (6), and our task is to determine the input value (x) that produces this output.

Types of Functions

Functions come in various forms, each with its unique characteristics and behavior. Recognizing the different types of functions is crucial for selecting the appropriate problem-solving techniques. Linear functions, such as h(x) = 5 - 2x, are characterized by a constant rate of change and can be represented by a straight line on a graph. They have the general form f(x) = mx + b, where m is the slope and b is the y-intercept. Quadratic functions have a squared term and form a parabola when graphed. They have the general form f(x) = ax² + bx + c. Polynomial functions are sums of terms involving variables raised to non-negative integer powers. Exponential functions involve a constant base raised to a variable exponent, exhibiting rapid growth or decay. Trigonometric functions relate angles of a triangle to the ratios of its sides, and they are periodic, repeating their values over regular intervals. Other types of functions include logarithmic, rational, and radical functions, each with its distinctive properties. The type of function involved in a problem can significantly influence the method used to find the input for a given output. For instance, solving for the input of a linear function is typically a straightforward algebraic process, while solving for the input of a quadratic function may involve factoring, completing the square, or using the quadratic formula. In our case, the function h(x) = 5 - 2x is a linear function, which simplifies the process of finding the input value.

Problem Statement: Finding the Input for h(x) = 6

Now, let's focus on the specific problem at hand. We are given the function h(x) = 5 - 2x, and we need to determine the input value x that produces an output of 6. In mathematical terms, we need to solve the equation h(x) = 6 for x. This means finding the value of x that, when plugged into the function, makes the expression 5 - 2x equal to 6. This type of problem is a common application of function concepts and requires a solid understanding of algebraic manipulation. It's like working backwards through the function machine: we know the output we want, and we need to figure out what input we need to feed into the machine to get that output. To solve this, we will use algebraic techniques to isolate x on one side of the equation. The steps involved will include substituting the output value into the function, simplifying the equation, and then performing operations to get x by itself. This process highlights the inverse relationship between input and output in a function, and it's a fundamental skill for working with mathematical models and relationships. The ability to find the input for a specific output is essential in many real-world applications, such as determining the amount of material needed to produce a certain quantity of goods or calculating the initial velocity required to reach a specific target distance.

Setting up the Equation

The first step in solving for the input x when h(x) = 6 is to set up the equation correctly. We know that h(x) is equal to 5 - 2x, and we are given that h(x) = 6. Therefore, we can substitute 6 for h(x) in the equation, which gives us the equation 6 = 5 - 2x. This equation represents the mathematical relationship we need to solve. It states that the output of the function, which is 6, is equal to the expression 5 - 2x, which depends on the input x. Setting up the equation correctly is crucial because it lays the foundation for the subsequent algebraic steps. A mistake in this initial step will lead to an incorrect solution. The equation 6 = 5 - 2x is a linear equation in one variable (x), which means it can be solved using basic algebraic operations. The goal is to isolate x on one side of the equation by performing the same operations on both sides to maintain the equality. This process will involve subtracting 5 from both sides and then dividing both sides by -2. The ability to set up equations correctly based on given information is a fundamental skill in mathematics and is essential for solving a wide range of problems, not just those involving functions.

Solving the Equation: A Step-by-Step Approach

With the equation 6 = 5 - 2x established, the next step is to solve for x. This involves using algebraic techniques to isolate x on one side of the equation. The key principle here is to perform the same operations on both sides of the equation to maintain the equality. This ensures that the solution we obtain is valid. We will follow a systematic approach, reversing the order of operations that are applied to x in the expression 5 - 2x. First, we will eliminate the constant term (5) by subtracting it from both sides. Then, we will eliminate the coefficient of x (-2) by dividing both sides by it. These steps will gradually simplify the equation until we have x by itself on one side, giving us the solution. Each step is crucial, and careful attention to detail is necessary to avoid errors. The process of solving equations is a cornerstone of algebra and is essential for solving many mathematical problems. It requires a clear understanding of algebraic properties and operations, as well as the ability to apply them in a logical and systematic way. In this case, solving the equation will reveal the input value x that corresponds to an output of 6 for the function h(x) = 5 - 2x.

Step 1: Subtracting 5 from Both Sides

The first step in isolating x in the equation 6 = 5 - 2x is to eliminate the constant term, which is 5. To do this, we subtract 5 from both sides of the equation. Subtracting the same value from both sides maintains the equality, ensuring that the equation remains balanced. This step is based on the subtraction property of equality, which states that if a = b, then a - c = b - c. Applying this property to our equation, we subtract 5 from both sides: 6 - 5 = 5 - 2x - 5. This simplifies to 1 = -2x. Now, the constant term on the right side of the equation has been eliminated, and we are left with a simpler equation that involves only the term with x. This step moves us closer to isolating x and finding its value. Subtracting constants from both sides of an equation is a common technique in algebra and is used to simplify equations and make them easier to solve. It's a fundamental skill that is applied in a wide range of mathematical contexts. In this specific case, subtracting 5 from both sides has effectively removed the constant term that was interfering with our ability to isolate x.

Step 2: Dividing Both Sides by -2

Having simplified the equation to 1 = -2x, the next step is to isolate x by eliminating its coefficient, which is -2. To do this, we divide both sides of the equation by -2. Dividing both sides by the same non-zero value maintains the equality, based on the division property of equality, which states that if a = b, then a / c = b / c (where c is not zero). Applying this property, we divide both sides of 1 = -2x by -2: 1 / -2 = -2x / -2. This simplifies to -1/2 = x. Now, x is completely isolated on one side of the equation, and we have found its value. This step completes the process of solving for x and provides the solution to our problem. Dividing both sides of an equation by the coefficient of the variable is a standard technique in algebra and is used to solve equations of the form ax = b. It's a crucial skill for finding the value of the unknown variable and is applied extensively in mathematics and related fields. In this case, dividing by -2 has revealed that the input value x that produces an output of 6 for the function h(x) = 5 - 2x is -1/2.

Solution: x = -1/2

After performing the algebraic steps of subtracting 5 from both sides and then dividing both sides by -2, we arrive at the solution x = -1/2. This means that when the input value x is -1/2, the output of the function h(x) = 5 - 2x is 6. This solution can be interpreted as a specific point on the graph of the function, where the x-coordinate is -1/2 and the y-coordinate (which represents the output h(x)) is 6. The solution x = -1/2 is a single, unique value because the function h(x) = 5 - 2x is a linear function, and linear functions have a one-to-one relationship between inputs and outputs (except for horizontal lines). This means that for every output value, there is only one corresponding input value. The solution x = -1/2 satisfies the original problem statement, which was to find the input x when the output h(x) is 6. It's a precise and accurate answer that can be verified by substituting it back into the original equation. The ability to solve for the input of a function given a specific output is a fundamental skill in mathematics and has applications in various fields, such as physics, engineering, and economics.

Verifying the Solution

To ensure the accuracy of our solution, it's always a good practice to verify it. Verification involves substituting the solution back into the original equation and checking if it holds true. In this case, our solution is x = -1/2, and the original equation is h(x) = 5 - 2x. We want to check if h(-1/2) is indeed equal to 6. Substituting x = -1/2 into the function, we get h(-1/2) = 5 - 2(-1/2). Simplifying this expression, we have h(-1/2) = 5 + 1 = 6. This confirms that our solution x = -1/2 is correct, as it produces the desired output of 6. Verification is an essential step in the problem-solving process because it helps to catch any errors that may have occurred during the solution process. It provides confidence in the accuracy of the answer and ensures that it satisfies the given conditions. In complex problems, verification can be particularly important, as it can help to identify subtle mistakes that may not be immediately obvious. The process of verification reinforces the understanding of the relationship between input and output in a function and highlights the importance of careful and accurate calculations. In this case, the successful verification of our solution demonstrates that we have correctly solved for the input value x that corresponds to the given output value of 6.

Conclusion

In conclusion, we have successfully determined the input value x for the function h(x) = 5 - 2x when the output is 6. By setting up the equation 6 = 5 - 2x and using algebraic techniques to isolate x, we found the solution to be x = -1/2. This solution was then verified by substituting it back into the original equation, confirming its accuracy. This process highlights the fundamental concept of functions and the relationship between input and output. It demonstrates the importance of algebraic manipulation in solving mathematical problems and the value of verification in ensuring the correctness of solutions. The ability to find the input for a given output is a crucial skill in mathematics and has applications in various fields. This step-by-step guide provides a clear and concise explanation of the process, making it accessible to learners of all levels. By understanding the concepts and techniques presented in this article, readers can confidently tackle similar problems and deepen their understanding of functions and their applications. The problem we solved is a simple example of a common type of problem in mathematics, and the techniques used can be applied to more complex functions and equations. The key is to understand the underlying principles and to apply them systematically and carefully.