Evaluating The Function F(x) = X^2 + 5x At X = P - 1

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Understanding Function Evaluation

In mathematics, a function is a rule that assigns to each input exactly one output. Evaluating a function means substituting a given value (or expression) for the variable in the function's formula and simplifying the result. This process allows us to determine the output of the function for a specific input. For the function at hand, f(x) = x^2 + 5x, we aim to evaluate it at x = p - 1. This involves replacing every instance of x in the function's expression with the expression (p - 1). This substitution is a fundamental concept in algebra and calculus, laying the groundwork for understanding more complex mathematical concepts.

To evaluate the function f(x) = x^2 + 5x at x = p - 1, we follow a systematic approach. The first step is direct substitution, where we replace each x in the function's formula with the expression (p - 1). This yields f(p - 1) = (p - 1)^2 + 5(p - 1). This initial substitution sets the stage for the simplification process, where we apply algebraic principles to arrive at a more concise expression. Understanding this substitution step is crucial, as it forms the basis for evaluating any function at a given value or expression. It is a versatile technique used extensively in various mathematical contexts, from solving equations to analyzing the behavior of functions.

Step-by-Step Evaluation of f(p - 1)

Now, let's delve into the simplification process. We start by expanding the squared term (p - 1)^2. This can be done using the binomial expansion formula, which states that (a - b)^2 = a^2 - 2ab + b^2. Applying this formula to our expression, we get (p - 1)^2 = p^2 - 2(p)(1) + 1^2 = p^2 - 2p + 1. This expansion is a critical step, as it transforms the squared term into a more manageable polynomial expression. Following this, we distribute the 5 in the term 5(p - 1), which gives us 5p - 5. This distribution simplifies the term and prepares it for combining with the expanded squared term. By performing these algebraic manipulations, we progressively simplify the expression, bringing us closer to the final evaluated form of the function.

With the expansion and distribution completed, we now have f(p - 1) = p^2 - 2p + 1 + 5p - 5. The next step is to combine like terms, which involves grouping together terms with the same variable and exponent. In this case, we have the p^2 term, the p terms (-2p and 5p), and the constant terms (1 and -5). Combining the p terms, we get -2p + 5p = 3p. Combining the constant terms, we get 1 - 5 = -4. Therefore, the simplified expression becomes f(p - 1) = p^2 + 3p - 4. This final expression represents the value of the function f(x) when evaluated at x = p - 1. The process of combining like terms is a fundamental algebraic skill that allows us to express polynomials in their simplest form, making them easier to analyze and work with. The ability to combine like terms efficiently is essential for solving equations, simplifying expressions, and performing various other mathematical operations.

Detailed Solution

Let's evaluate the function step-by-step:

  1. Substitute x with (p - 1) in the function: f(p - 1) = (p - 1)^2 + 5(p - 1)
  2. Expand the squared term: (p - 1)^2 = p^2 - 2p + 1
  3. Distribute the 5: 5(p - 1) = 5p - 5
  4. Substitute the expanded terms back into the expression: f(p - 1) = p^2 - 2p + 1 + 5p - 5
  5. Combine like terms: f(p - 1) = p^2 + (-2p + 5p) + (1 - 5)
  6. Simplify: f(p - 1) = p^2 + 3p - 4

Therefore, f(p - 1) = p^2 + 3p - 4 is the simplified answer.

Alternative Methods for Expansion and Simplification

While the binomial expansion formula is a standard method for expanding squared terms, alternative approaches can sometimes be more intuitive or efficient. For example, we could use the FOIL (First, Outer, Inner, Last) method to expand (p - 1)^2. This method involves multiplying each term in the first binomial by each term in the second binomial: (p - 1)(p - 1) = (p * p) + (p * -1) + (-1 * p) + (-1 * -1) = p^2 - p - p + 1 = p^2 - 2p + 1. This approach can be particularly useful for individuals who prefer a more visual and step-by-step method. Additionally, understanding multiple methods for expansion and simplification can enhance problem-solving flexibility and provide a deeper understanding of algebraic principles.

Another aspect of simplification involves recognizing patterns and applying algebraic identities. In some cases, expressions can be simplified more readily by identifying common factors or using identities such as the difference of squares or the sum/difference of cubes. These identities provide shortcuts for simplifying certain types of expressions, making the process more efficient. In the context of our function evaluation, there are no direct applications of such identities. However, in more complex problems, recognizing and utilizing these patterns can significantly reduce the computational effort required for simplification. Therefore, developing proficiency in recognizing and applying algebraic identities is a valuable skill in mathematics.

Common Mistakes and How to Avoid Them

Evaluating functions and simplifying expressions are fundamental skills in algebra, but they are also areas where students commonly make mistakes. One frequent error is incorrect substitution, where the given value or expression is not properly substituted for the variable in the function's formula. To avoid this, it is crucial to carefully replace each instance of the variable with the given value, paying close attention to parentheses and signs. For example, when evaluating f(x) = x^2 + 5x at x = p - 1, it is essential to substitute (p - 1) for x throughout the expression, ensuring that the entire expression (p - 1) is squared and multiplied by 5. Neglecting parentheses or misinterpreting the order of operations can lead to incorrect substitutions and, consequently, incorrect results. Therefore, paying meticulous attention to detail during the substitution process is paramount.

Another common mistake occurs during the expansion of squared terms. Students may incorrectly apply the distributive property or forget to include all the necessary terms. For instance, when expanding (p - 1)^2, it is essential to remember that this expression means (p - 1)(p - 1), and the FOIL method or the binomial expansion formula must be applied correctly. A frequent error is writing (p - 1)^2 as p^2 - 1, which neglects the middle term -2p. To avoid this, it is helpful to write out the binomial multiplication explicitly and carefully apply the distributive property. Furthermore, understanding the binomial expansion formula and its derivation can reinforce the correct procedure for expanding squared terms. Regular practice and careful attention to detail can significantly reduce the likelihood of making errors in expansion.

Errors also often arise when combining like terms. Students may incorrectly add or subtract coefficients or combine terms with different variables or exponents. To minimize these errors, it is helpful to organize terms systematically and group like terms together before performing any operations. For example, in the expression p^2 - 2p + 1 + 5p - 5, one can rearrange the terms as p^2 + (-2p + 5p) + (1 - 5) to clearly identify the like terms. Additionally, double-checking the signs and coefficients before combining terms can prevent careless mistakes. A strong understanding of algebraic principles and attention to detail are crucial for accurate simplification.

Conclusion

In summary, evaluating the function f(x) = x^2 + 5x at x = p - 1 involves careful substitution, expansion, and simplification. The process requires a solid understanding of algebraic principles and attention to detail. The final simplified expression, f(p - 1) = p^2 + 3p - 4, represents the value of the function at the given input. By mastering these techniques, students can confidently tackle more complex mathematical problems and develop a deeper appreciation for the power and elegance of algebra.

This evaluation process showcases the fundamental concept of function evaluation, which is crucial in various areas of mathematics. From solving equations to analyzing the behavior of functions, the ability to substitute and simplify expressions is essential. This example provides a clear illustration of how these skills are applied in practice, reinforcing the importance of mastering algebraic techniques. Through consistent practice and attention to detail, students can develop proficiency in evaluating functions and simplifying expressions, paving the way for success in more advanced mathematical studies.