Evaluating F(x+h) For F(x) = X² + 7x A Step-by-Step Guide
In mathematics, functions are fundamental tools for describing relationships between variables. Evaluating a function at a specific value is a crucial skill. It allows us to understand the function's behavior and its output for a given input. This article will delve into evaluating the function f(x) = x² + 7x at x + h. This is a common operation in calculus, particularly when calculating derivatives using the limit definition. By understanding this process, we lay the groundwork for more advanced mathematical concepts. The ability to manipulate algebraic expressions and simplify them is paramount in mathematics. In this context, we'll explore how to substitute an expression (x + h) into a function and then simplify the resulting expression. This involves applying the distributive property, expanding squared terms, and combining like terms. These skills are essential not only for evaluating functions but also for various algebraic manipulations and problem-solving scenarios. Understanding the mechanics of function evaluation, especially with algebraic expressions as inputs, is crucial for grasping concepts like the difference quotient, which forms the basis for derivatives in calculus. This article will provide a step-by-step approach to evaluating f(x + h), emphasizing the underlying principles of algebraic manipulation and simplification. This knowledge will be invaluable for students venturing into calculus and other advanced mathematical topics. By mastering the techniques presented here, readers will be well-equipped to tackle more complex function evaluations and algebraic problems.
Understanding the Function f(x) = x² + 7x
Before we jump into evaluating f(x + h), let's first understand the function f(x) = x² + 7x itself. This is a quadratic function, which means it has a term with x². Quadratic functions are characterized by their parabolic shape when graphed. The x² term dictates the overall curvature of the parabola, while the 7x term influences its position and orientation in the coordinate plane. To further dissect this function, we can identify its key components. The coefficient of the x² term is 1, which indicates that the parabola opens upwards. The coefficient of the x term is 7, which affects the parabola's axis of symmetry and vertex. The absence of a constant term implies that the parabola passes through the origin (0, 0). Understanding the characteristics of quadratic functions is essential for various applications, including optimization problems, projectile motion analysis, and curve fitting. By analyzing the coefficients and terms, we can predict the function's behavior and graph. Moreover, familiarity with quadratic functions lays the foundation for more complex polynomial functions. Evaluating a function at specific points provides valuable insights into its behavior. For example, evaluating f(x) at x = 0, x = 1, and x = -1 gives us f(0) = 0, f(1) = 8, and f(-1) = -6, respectively. These points can help us sketch the graph of the parabola and understand its increasing and decreasing intervals. Furthermore, understanding the function's behavior allows us to predict its output for different input values and to solve equations involving the function. The function f(x) = x² + 7x can also be viewed in the context of transformations of the basic quadratic function y = x². The 7x term represents a horizontal shift and a vertical stretch of the basic parabola. Understanding these transformations allows us to visualize the graph of f(x) more easily and to compare it with other quadratic functions. In summary, a thorough understanding of the function f(x) = x² + 7x is crucial for evaluating it at any input, including x + h. By analyzing its components, characteristics, and behavior, we can approach the evaluation process with confidence and accuracy.
Substituting x + h into f(x)
Now that we have a good understanding of the function f(x) = x² + 7x, let's move on to the core of the problem: evaluating f(x + h). This means we need to replace every instance of x in the function's expression with the expression (x + h). This process is known as substitution and is a fundamental technique in algebra and calculus. Substituting x + h into f(x) = x² + 7x gives us f(x + h) = (x + h)² + 7(x + h). Notice that we've replaced x with (x + h) in both the x² term and the 7x term. It's crucial to use parentheses around (x + h) to ensure that the entire expression is treated as a single entity. This is particularly important when dealing with squared terms and distributive properties. The next step involves expanding and simplifying the expression we obtained after substitution. This requires applying algebraic rules such as the distributive property and the formula for squaring a binomial. Expanding and simplifying is essential to obtain a clear and manageable expression for f(x + h). Failing to do so can lead to errors in subsequent calculations, especially when using f(x + h) in more complex operations like finding derivatives. The expression (x + h)² represents the square of a binomial, which can be expanded using the formula (a + b)² = a² + 2ab + b². In our case, this translates to (x + h)² = x² + 2xh + h². This expansion is a fundamental algebraic identity that's used extensively in mathematics. It's crucial to memorize and understand this formula to efficiently expand squared binomials. The term 7(x + h) can be expanded using the distributive property, which states that a(b + c) = ab + ac. Applying this property, we get 7(x + h) = 7x + 7h. The distributive property is another fundamental algebraic rule that's used to multiply a term by a sum or difference. By correctly applying these algebraic rules, we can expand the expression for f(x + h) and prepare it for simplification. The expanded form of f(x + h) will then be x² + 2xh + h² + 7x + 7h. This expression contains several terms, some of which may be like terms that can be combined to further simplify the expression. The next step in the evaluation process involves identifying and combining these like terms.
Expanding and Simplifying the Expression
After substituting x + h into f(x), we arrived at the expression f(x + h) = (x + h)² + 7(x + h). The next crucial step is to expand and simplify this expression. This involves applying algebraic rules and techniques to obtain a more manageable form. Expanding the expression means removing the parentheses by performing the indicated operations. This typically involves using the distributive property and the formula for squaring a binomial. The goal of simplification is to combine like terms and reduce the expression to its simplest form. This makes it easier to work with in subsequent calculations and analysis. To expand (x + h)², we use the formula (a + b)² = a² + 2ab + b². Applying this formula, we get (x + h)² = x² + 2xh + h². This expansion is a fundamental algebraic identity and is essential for simplifying expressions involving squared binomials. The term 2xh represents twice the product of x and h, and h² represents the square of h. These terms are crucial for understanding the behavior of the function as h changes. Next, we expand 7(x + h) using the distributive property, which states that a(b + c) = ab + ac. Applying this property, we get 7(x + h) = 7x + 7h. This step involves multiplying 7 by both x and h inside the parentheses. The distributive property is a fundamental algebraic rule that's used extensively in simplifying expressions. Combining the expanded terms, we have f(x + h) = x² + 2xh + h² + 7x + 7h. Now, we look for like terms that can be combined. Like terms are terms that have the same variable raised to the same power. In this expression, there are no like terms that can be combined. The terms x², 2xh, h², 7x, and 7h are all distinct and cannot be further simplified. Therefore, the simplified expression for f(x + h) is x² + 2xh + h² + 7x + 7h. This expression is the final result of evaluating the function f(x) = x² + 7x at x + h. It represents the output of the function when the input is x + h. This simplified expression is often used in calculus to calculate the derivative of the function using the limit definition.
Final Result: f(x + h) = x² + 2xh + h² + 7x + 7h
After carefully substituting x + h into the function f(x) = x² + 7x, expanding the terms, and simplifying the expression, we have arrived at the final result: f(x + h) = x² + 2xh + h² + 7x + 7h. This expression represents the value of the function f when the input is x + h. It's a crucial result that has significant applications in calculus and other areas of mathematics. The expression consists of five terms: x², 2xh, h², 7x, and 7h. Each term plays a specific role in determining the value of f(x + h). The x² term represents the square of the original variable x. The 2xh term represents twice the product of x and the increment h. The h² term represents the square of the increment h. The 7x term represents seven times the original variable x, and the 7h term represents seven times the increment h. This expression is often used in the definition of the derivative of a function. The derivative measures the instantaneous rate of change of a function with respect to its input variable. It's a fundamental concept in calculus and has numerous applications in physics, engineering, economics, and other fields. The expression for f(x + h) is a key component in the difference quotient, which is used to calculate the derivative. The difference quotient is defined as (f(x + h) - f(x)) / h. By substituting the expression for f(x + h) and simplifying, we can obtain an expression for the difference quotient. Taking the limit of the difference quotient as h approaches 0 gives us the derivative of the function. The result f(x + h) = x² + 2xh + h² + 7x + 7h also provides insights into the behavior of the function f(x) = x² + 7x. It shows how the output of the function changes as the input changes from x to x + h. The terms involving h represent the change in the function's value due to the increment h. Understanding this expression is crucial for comprehending the concept of function evaluation and its applications in calculus and other areas of mathematics. It's a fundamental building block for more advanced mathematical concepts and techniques. By mastering the process of evaluating functions at algebraic expressions, students can develop a deeper understanding of mathematical relationships and problem-solving strategies.
In conclusion, evaluating the function f(x) = x² + 7x at x + h involves substituting x + h for x in the function's expression, expanding the resulting expression, and simplifying it by combining like terms. This process yields the result f(x + h) = x² + 2xh + h² + 7x + 7h. This skill is fundamental in mathematics, particularly in calculus, where it's used to calculate derivatives using the limit definition. Understanding how to manipulate algebraic expressions and simplify them is crucial for various mathematical operations and problem-solving scenarios. The ability to evaluate functions at algebraic expressions, like x + h, is essential for grasping concepts such as the difference quotient and derivatives. This article provided a step-by-step guide to evaluating f(x + h), emphasizing the underlying principles of algebraic manipulation and simplification. By mastering these techniques, readers can confidently tackle more complex function evaluations and algebraic problems. The result f(x + h) = x² + 2xh + h² + 7x + 7h is not just a final answer; it's a building block for further mathematical exploration. It allows us to analyze the function's behavior, calculate its derivatives, and solve related problems. The process of evaluating functions at algebraic expressions reinforces fundamental algebraic skills, such as the distributive property and the formula for squaring a binomial. These skills are essential for success in mathematics and related fields. Furthermore, the concept of function evaluation is not limited to quadratic functions like f(x) = x² + 7x. It applies to a wide range of functions, including polynomial, trigonometric, exponential, and logarithmic functions. The techniques learned in this article can be extended to evaluate any function at any algebraic expression. Ultimately, the ability to evaluate functions and manipulate algebraic expressions is a cornerstone of mathematical proficiency. It empowers students to understand and apply mathematical concepts in various contexts. By mastering these skills, readers can unlock a deeper appreciation for the power and beauty of mathematics. This article has provided a comprehensive guide to evaluating the function f(x) = x² + 7x at x + h, equipping readers with the knowledge and skills to tackle similar problems and explore more advanced mathematical topics.
