Given \[$ F(x) \$\], Evaluate \[$ F(2) \$\].$\[ \begin{array}{c} f(x)=\frac{3x^2-x}{5x} \\ f(2)= \, [?] \end{array} \\]

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Introduction

In mathematics, functions are a fundamental concept that helps us describe relationships between variables. Evaluating a function at a specific value of its variable is a crucial operation that allows us to determine the output of the function for a given input. In this article, we will focus on evaluating the function f(x) = (3x^2 - x) / 5x at x = 2.

Understanding the Function

Before we can evaluate the function, we need to understand its structure. The function f(x) is defined as:

f(x) = (3x^2 - x) / 5x

This function takes a value of x as input and returns a value as output. The function consists of two main parts: the numerator (3x^2 - x) and the denominator (5x).

Evaluating the Function at x = 2

Now that we have a clear understanding of the function, we can proceed to evaluate it at x = 2. To do this, we need to substitute x = 2 into the function and simplify the expression.

f(2) = (3(2)^2 - 2) / 5(2)

Step 1: Evaluate the Exponents

The first step is to evaluate the exponents in the expression. In this case, we have (2)^2, which is equal to 4.

f(2) = (3(4) - 2) / 5(2)

Step 2: Multiply the Numbers

Next, we need to multiply the numbers in the expression. We have 3(4), which is equal to 12, and 5(2), which is equal to 10.

f(2) = (12 - 2) / 10

Step 3: Subtract the Numbers

Now, we need to subtract 2 from 12.

f(2) = 10 / 10

Step 4: Simplify the Expression

Finally, we can simplify the expression by dividing 10 by 10, which is equal to 1.

f(2) = 1

Conclusion

In this article, we evaluated the function f(x) = (3x^2 - x) / 5x at x = 2. We followed a step-by-step approach to simplify the expression and arrived at the final answer, which is f(2) = 1. This example demonstrates the importance of understanding the structure of a function and the need to follow a systematic approach when evaluating it at a specific value of its variable.

Common Mistakes to Avoid

When evaluating a function, it's essential to avoid common mistakes that can lead to incorrect results. Some of these mistakes include:

  • Not following the order of operations: When evaluating a function, it's crucial to follow the order of operations (PEMDAS) to ensure that the expression is simplified correctly.
  • Not simplifying the expression: Failing to simplify the expression can lead to incorrect results. Make sure to simplify the expression by combining like terms and canceling out common factors.
  • Not checking for domain restrictions: Some functions have domain restrictions that need to be considered when evaluating them. Make sure to check the domain restrictions of the function before evaluating it.

Real-World Applications

Evaluating functions has numerous real-world applications in various fields, including:

  • Physics: Evaluating functions is essential in physics to describe the motion of objects and predict their behavior under different conditions.
  • Engineering: Evaluating functions is crucial in engineering to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Evaluating functions is essential in economics to model economic systems and predict the behavior of economic variables.

Conclusion

Introduction

In our previous article, we discussed how to evaluate the function f(x) = (3x^2 - x) / 5x at x = 2. In this article, we will provide a Q&A guide to help you understand the concept of evaluating functions and how to apply it in different scenarios.

Q: What is a function?

A: A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). It is a way of describing a relationship between variables.

Q: What is the purpose of evaluating a function?

A: The purpose of evaluating a function is to determine the output of the function for a given input. This is useful in a variety of applications, such as physics, engineering, and economics.

Q: How do I evaluate a function?

A: To evaluate a function, you need to follow these steps:

  1. Substitute the input value: Replace the variable (x) with the given input value.
  2. Simplify the expression: Use the order of operations (PEMDAS) to simplify the expression.
  3. Check for domain restrictions: Make sure the input value is within the domain of the function.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells you which operations to perform first when evaluating an expression. The acronym PEMDAS stands for:

  1. P: Parentheses (evaluate expressions inside parentheses first)
  2. E: Exponents (evaluate any exponential expressions next)
  3. M: Multiplication and Division (evaluate multiplication and division operations from left to right)
  4. A: Addition and Subtraction (evaluate any addition and subtraction operations from left to right)

Q: How do I simplify an expression?

A: To simplify an expression, you need to:

  1. Combine like terms: Combine any terms that have the same variable and exponent.
  2. Cancel out common factors: Cancel out any common factors between the numerator and denominator.
  3. Simplify fractions: Simplify any fractions by dividing the numerator and denominator by their greatest common divisor.

Q: What are some common mistakes to avoid when evaluating a function?

A: Some common mistakes to avoid when evaluating a function include:

  • Not following the order of operations: Failing to follow the order of operations (PEMDAS) can lead to incorrect results.
  • Not simplifying the expression: Failing to simplify the expression can lead to incorrect results.
  • Not checking for domain restrictions: Failing to check for domain restrictions can lead to incorrect results.

Q: How do I check for domain restrictions?

A: To check for domain restrictions, you need to:

  1. Check for division by zero: Make sure the denominator is not equal to zero.
  2. Check for square roots of negative numbers: Make sure the expression under the square root is not negative.
  3. Check for other restrictions: Check for any other restrictions on the domain of the function.

Conclusion

In conclusion, evaluating functions is a fundamental operation in mathematics that has numerous real-world applications. By following a step-by-step approach and avoiding common mistakes, we can ensure that we arrive at the correct result. Whether you're a student, a professional, or simply someone interested in mathematics, understanding how to evaluate functions is an essential skill that can benefit you in many ways.