Evaluate Function F(x) = X^2 + 5x At F(x-3)

In mathematics, a function is a relationship between a set of inputs and a set of possible outputs, with the property that each input is related to exactly one output. Function evaluation is the process of finding the output value of a function for a specific input value. This involves substituting the given input value into the function's expression and simplifying the result. Understanding function evaluation is crucial for various mathematical concepts and applications, including calculus, algebra, and data analysis.

To effectively evaluate functions, one must grasp the fundamental principles of algebraic manipulation and substitution. The core concept revolves around replacing the variable in the function's expression with the provided input value. This input value could be a numerical constant, another variable, or even a more complex algebraic expression. The subsequent steps involve simplifying the expression by applying the order of operations (PEMDAS/BODMAS), which encompasses parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right). A firm understanding of these rules is essential for accurate function evaluation.

Function evaluation extends beyond simple numerical substitutions. Functions can be evaluated at algebraic expressions, which means substituting an expression containing variables for the function's input variable. This technique is particularly useful in various mathematical contexts, such as finding composite functions, determining the inverse of a function, and exploring functional relationships. The process of evaluating at an expression involves similar steps to evaluating at a numerical value, but with the added complexity of manipulating algebraic terms. Careful attention to detail and a thorough understanding of algebraic principles are crucial for successful evaluation at algebraic expressions. It is also important to recognize that some functions may have restricted domains, meaning they are not defined for all input values. These restrictions can arise from various factors, such as division by zero, taking the square root of a negative number, or the presence of logarithms. When evaluating a function, it is essential to check whether the input value or expression falls within the function's domain. If the input is outside the domain, the function is not defined at that point, and the evaluation is not possible.

We are given the function:

$f(x) = x^2 + 5x$

We are asked to find the value of the function when the input is $(x - 3)$. In other words, we need to find $f(x - 3)$. This requires us to substitute $(x - 3)$ for $x$ in the expression for $f(x)$. The process involves careful algebraic manipulation and simplification to arrive at the final expression for $f(x - 3)$. This type of problem is a fundamental exercise in function evaluation, which is a critical skill in mathematics. Accurately evaluating functions at given expressions is essential for solving more complex problems in algebra, calculus, and other mathematical fields. It also reinforces the understanding of functions as mappings between inputs and outputs, where the input can be a variable, a number, or an algebraic expression. The ability to confidently perform function evaluation is a cornerstone of mathematical proficiency.

To evaluate $f(x - 3)$, we will substitute $(x - 3)$ for every instance of $x$ in the original function $f(x) = x^2 + 5x$. Here's the breakdown:

1. Substitution:

   Replace $x$ with $(x - 3)$ in the function:

   $f(x - 3) = (x - 3)^2 + 5(x - 3)$

   This step directly applies the definition of function evaluation, where we replace the variable with the given expression. It's crucial to use parentheses to ensure that the entire expression $(x - 3)$ is treated as a single entity when applying the function's operations. Neglecting the parentheses can lead to incorrect results, especially when dealing with exponents or multiplication.

2. Expand the squared term:

   Expand $(x - 3)^2$ using the FOIL method or the binomial square formula $((a - b)^2 = a^2 - 2ab + b^2)$:

   $(x - 3)^2 = x^2 - 2(x)(3) + 3^2 = x^2 - 6x + 9$

   Expanding the squared term is a critical step in simplifying the expression. The binomial square formula provides a direct and efficient way to perform this expansion. The FOIL method (First, Outer, Inner, Last) can also be used, but it is essentially the same process. The key is to correctly apply the distributive property to multiply the binomial by itself. A common error is to simply square each term individually, which would result in $x^2 + 9$ instead of the correct expansion.

3. Distribute the 5:

   Distribute the $5$ in the term $5(x - 3)$:

   $5(x - 3) = 5x - 15$

   The distributive property states that $a(b + c) = ab + ac$. In this step, we apply this property to multiply the constant $5$ by each term inside the parentheses. This is a straightforward application of the distributive property, but it's essential to perform it correctly to maintain the accuracy of the expression. Errors in distribution can significantly alter the result of the evaluation.

4. Combine like terms:

   Now, substitute the expanded terms back into the expression for $f(x - 3)$ and combine like terms:

   $f(x - 3) = (x^2 - 6x + 9) + (5x - 15)$

   $f(x - 3) = x^2 - 6x + 5x + 9 - 15$

   $f(x - 3) = x^2 - x - 6$

   Combining like terms involves identifying terms with the same variable and exponent and then adding or subtracting their coefficients. This step simplifies the expression by reducing the number of terms. In this case, we combine the $-6x$ and $5x$ terms to get $-x$, and we combine the constants $9$ and $-15$ to get $-6$. Careful attention to the signs of the coefficients is crucial for accurate simplification.

Therefore, $f(x - 3) = x^2 - x - 6$.

The final answer is:

$f(x - 3) = x^2 - x - 6$

This result represents the value of the function $f(x)$ when the input is the expression $(x - 3)$. The step-by-step solution demonstrates the process of function evaluation, which involves substituting the input expression into the function and simplifying the resulting expression using algebraic techniques. The final expression, $x^2 - x - 6$, is a quadratic expression that represents the transformed function. This quadratic expression can be further analyzed to find its roots, vertex, and other properties. The ability to perform function evaluation and algebraic manipulation is fundamental to understanding and working with functions in mathematics and other related fields.

Function Evaluation

Function evaluation is the process of determining the output value of a function for a given input value. This involves substituting the input value into the function's expression and simplifying the result. Function evaluation is a fundamental concept in mathematics, used extensively in various areas such as algebra, calculus, and analysis.

Substitution

Substitution is the process of replacing a variable in an expression with a specific value or another expression. In function evaluation, substitution is used to replace the input variable with the given input value or expression. Accurate substitution is crucial for correct function evaluation.

Order of Operations (PEMDAS/BODMAS)

The order of operations, often remembered by the acronyms PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction), dictates the sequence in which operations should be performed in a mathematical expression. Following the order of operations is essential for obtaining the correct result when simplifying expressions, particularly in function evaluation.

Algebraic Manipulation

Algebraic manipulation involves using algebraic rules and properties to simplify or transform expressions. This includes techniques such as expanding expressions, combining like terms, factoring, and simplifying fractions. Algebraic manipulation is a key skill in function evaluation, especially when dealing with complex expressions or functions defined by algebraic formulas.

Expanding Expressions

Expanding expressions involves removing parentheses or brackets by applying the distributive property or other algebraic rules. For example, expanding $(x - 3)^2$ involves applying the binomial square formula or the FOIL method. Correct expansion is crucial for simplifying expressions and performing accurate function evaluation.

Combining Like Terms

Combining like terms involves identifying terms in an expression that have the same variable and exponent and then adding or subtracting their coefficients. This simplifies the expression by reducing the number of terms. For example, in the expression $x^2 - 6x + 5x + 9 - 15$, the terms $-6x$ and $5x$ can be combined, as well as the constants $9$ and $-15$. Accurate combination of like terms is essential for simplifying expressions in function evaluation.

Function evaluation is a core concept in mathematics with numerous practical applications across various fields. Its importance stems from the fact that functions are used to model real-world phenomena and relationships. By evaluating functions, we can make predictions, analyze data, and solve problems in diverse areas such as physics, engineering, economics, and computer science.

In physics, functions are used to describe the motion of objects, the behavior of electromagnetic fields, and the properties of quantum systems. Evaluating these functions at specific times or positions allows physicists to predict the state of a system or the outcome of an experiment. For instance, the trajectory of a projectile can be modeled using a function, and evaluating this function at different times allows us to determine the projectile's position and velocity.

Engineering relies heavily on function evaluation for design, analysis, and optimization. Engineers use functions to model the behavior of circuits, the stress on structures, and the flow of fluids. Evaluating these functions helps engineers ensure the safety and efficiency of their designs. For example, structural engineers use functions to model the stress distribution in a bridge, and evaluating these functions helps them determine the bridge's load-bearing capacity.

In economics, functions are used to model market behavior, consumer demand, and economic growth. Evaluating these functions allows economists to make predictions about economic trends and the impact of policy decisions. For example, a demand function can be used to model the relationship between the price of a product and the quantity demanded, and evaluating this function at different prices allows economists to estimate the market demand for the product.

Computer science also utilizes function evaluation extensively in algorithm design, data analysis, and machine learning. Functions are the fundamental building blocks of computer programs, and evaluating functions is essential for executing code and producing results. In machine learning, functions are used to model relationships between data points, and evaluating these functions allows computers to make predictions and classify data. For example, a machine learning model might use a function to predict whether an email is spam based on its content, and evaluating this function for a new email allows the model to classify it as spam or not spam.

Beyond these specific fields, function evaluation is also a fundamental skill in data analysis, where it is used to interpret data, create visualizations, and build models. In statistics, function evaluation is used to calculate probabilities, test hypotheses, and make inferences. In general, function evaluation is a versatile tool that empowers individuals to understand and interact with the world around them. It allows us to quantify relationships, make predictions, and solve problems in a wide range of contexts.

When evaluating functions, several common mistakes can lead to incorrect results. Being aware of these pitfalls can help you avoid them and improve your accuracy.

One of the most frequent errors is incorrect substitution. This occurs when the input value or expression is not properly substituted into the function's expression. For example, when evaluating $f(x - 3)$ for the function $f(x) = x^2 + 5x$, a common mistake is to write $(x - 3^2) + 5(x - 3)$ instead of $(x - 3)^2 + 5(x - 3)$. The missing parentheses around $(x - 3)$ in the first term lead to an incorrect expansion. To avoid this, always ensure that the entire input expression is substituted for the variable, using parentheses when necessary.

Another common mistake is incorrectly applying the order of operations. As mentioned earlier, the order of operations (PEMDAS/BODMAS) dictates the sequence in which operations should be performed. Failing to follow this order can lead to errors in simplification. For instance, when evaluating an expression like $(x - 3)^2$, the exponent should be applied before any other operations. A common mistake is to distribute the square before dealing with the subtraction, which would lead to an incorrect result. To avoid this, always adhere to the order of operations when simplifying expressions.

Errors in algebraic manipulation are also a frequent source of mistakes. This includes errors in expanding expressions, combining like terms, and simplifying fractions. For example, when expanding $(x - 3)^2$, a common mistake is to write $x^2 - 9$ instead of the correct expansion $x^2 - 6x + 9$. To avoid these errors, review basic algebraic rules and practice algebraic manipulation techniques. Pay close attention to signs and coefficients when combining like terms, and double-check your work for accuracy.

Finally, not simplifying the expression completely can also be a mistake. Function evaluation typically involves simplifying the resulting expression as much as possible. Leaving an expression in an unsimplified form can make it difficult to interpret the result or use it in further calculations. Always simplify the expression by combining like terms and factoring where possible. By avoiding these common mistakes, you can improve your accuracy and confidence in evaluating functions.