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

Jul 6, 2025 by Admin 48 views

Evaluate the Function at the Given Expression

In the realm of mathematics, functions serve as fundamental building blocks, depicting relationships between inputs and outputs. Evaluating a function at a specific expression involves substituting the given expression for the function's variable and simplifying the resulting expression. This process allows us to determine the output of the function for a particular input, providing valuable insights into the function's behavior and characteristics. In this article, we will delve into the process of evaluating functions, using the example of the function $f(x) = x^2 + 5x$ and evaluating it at the expression $x - 3$ .

Before we embark on the evaluation process, it's crucial to grasp the essence of function notation. A function is typically represented as $f(x)$ , where $f$ is the function's name and $x$ signifies the input variable. The expression $f(x)$ denotes the output of the function when the input is $x$ . For instance, in the function $f(x) = x^2 + 5x$ , the output is obtained by squaring the input $x$ and adding it to five times the input $x$ .

Now, let's proceed with evaluating the function $f(x) = x^2 + 5x$ at the expression $x - 3$ . This entails substituting $x - 3$ for $x$ in the function's expression. Here's a breakdown of the steps involved:

Substitution: Replace every instance of $x$ in the function's expression with the expression $x - 3$ . This yields:

$f(x - 3) = (x - 3)^2 + 5(x - 3)$
Expansion: Expand the squared term and distribute the 5:

$f(x - 3) = (x^2 - 6x + 9) + (5x - 15)$
Simplification: Combine like terms to simplify the expression:

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

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

Therefore, the function $f(x) = x^2 + 5x$ evaluated at the expression $x - 3$ is $f(x - 3) = x^2 - x - 6$ .

To ensure the accuracy of our evaluation, we can employ a simple verification method. Let's choose a specific value for $x$ , such as $x = 4$ , and evaluate both the original function $f(x)$ and the evaluated function $f(x - 3)$ at this value.

Original Function:

$f(4) = 4^2 + 5(4) = 16 + 20 = 36$
Evaluated Function:

$f(4 - 3) = f(1) = 1^2 - 1 - 6 = 1 - 1 - 6 = -6$

Now, let's evaluate $f(x - 3) = x^2 - x - 6$ at $x = 4$ :

$f(4 - 3) = 4^2 - 4 - 6 = 16 - 4 - 6 = 6$

Since both evaluations yield the same result, our evaluation of the function at the given expression is verified.

Function evaluation is not merely an abstract mathematical exercise; it holds significant practical applications across diverse fields. Here are a few notable examples:

Physics: In physics, functions are used to model various physical phenomena, such as the motion of objects or the behavior of electric circuits. Evaluating these functions at specific values allows physicists to predict the outcome of experiments or analyze real-world scenarios.
Engineering: Engineers utilize functions to design and analyze structures, systems, and processes. Evaluating these functions helps engineers determine the performance of a design under different conditions or optimize the design for specific requirements.
Computer Science: Computer scientists heavily rely on functions in programming. Evaluating functions is essential for executing code, testing software, and developing algorithms.
Economics: Economists employ functions to model economic trends and predict market behavior. Evaluating these functions allows economists to analyze the impact of different policies or forecast economic growth.

To ensure accuracy and efficiency when evaluating functions, consider these helpful tips:

Pay close attention to notation: Ensure you understand the function's notation and what each variable represents.
Substitute carefully: When substituting the expression for the variable, double-check that you've replaced every instance of the variable.
Simplify systematically: Expand and simplify the expression step by step, combining like terms to avoid errors.
Verify your result: Use a specific value for the variable to verify that your evaluated function produces the same output as the original function.

Let's reinforce our understanding of function evaluation with a few more examples:

Example 1:

Given the function $g(x) = 3x^2 - 2x + 1$ , evaluate $g(2x)$ .

Substitution: Replace $x$ with $2x$ :

$g(2x) = 3(2x)^2 - 2(2x) + 1$
Expansion: Simplify the expression:

$g(2x) = 3(4x^2) - 4x + 1$
Simplification: Combine like terms:

$g(2x) = 12x^2 - 4x + 1$

Example 2:

Given the function $h(t) = rac{t + 1}{t - 1}$ , evaluate $h(t + 1)$ .

Substitution: Replace $t$ with $t + 1$ :

$h(t + 1) = rac{(t + 1) + 1}{(t + 1) - 1}$
Simplification: Simplify the expression:

$h(t + 1) = rac{t + 2}{t}$

While function evaluation is a straightforward process, certain pitfalls can lead to errors. Here are some common mistakes to watch out for:

Incorrect Substitution: Ensure you substitute the expression correctly for every instance of the variable.
Order of Operations: Adhere to the order of operations (PEMDAS/BODMAS) when simplifying the expression.
Sign Errors: Pay close attention to signs, especially when dealing with negative numbers or subtractions.
Algebraic Mistakes: Double-check your algebraic manipulations, such as expanding and combining like terms.

Evaluating functions at given expressions is a fundamental skill in mathematics with wide-ranging applications. By following the steps outlined in this article, you can confidently evaluate functions and gain a deeper understanding of their behavior. Remember to pay close attention to notation, substitute carefully, simplify systematically, and verify your results. With practice, you'll master this essential skill and unlock the power of functions in various mathematical and real-world contexts.

Through function evaluation, we bridge the gap between abstract mathematical concepts and concrete applications. Whether it's predicting the trajectory of a projectile, optimizing a chemical process, or analyzing economic trends, function evaluation empowers us to make informed decisions and solve complex problems. So, embrace the power of function evaluation and unlock its potential to enhance your understanding of the world around you.

By mastering the art of evaluating functions, you equip yourself with a valuable tool that transcends the boundaries of mathematics and extends into various domains of human endeavor. Whether you're a student delving into the intricacies of calculus or a professional applying mathematical models in your field, the ability to evaluate functions with precision and confidence will undoubtedly serve you well.

So, let's embark on this journey of mathematical exploration, where functions serve as our compass and evaluation as our guide. Together, we'll unravel the secrets hidden within these elegant mathematical constructs and unlock their potential to shape our understanding of the world.