Consider The Function $ F(x) = \frac{x+1}{x^2 - 5x + 6} $Evaluate $ F(x-1) = \square $

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Introduction

In this article, we will explore the concept of function evaluation, specifically focusing on the function $ f(x) = \frac{x+1}{x^2 - 5x + 6} $. We will evaluate the function at $ x-1 $, which involves substituting $ x-1 $ into the function in place of $ x $. This process will help us understand how to manipulate and simplify algebraic expressions.

Understanding the Function

The given function is $ f(x) = \frac{x+1}{x^2 - 5x + 6} $. To evaluate this function, we need to understand its components. The numerator is $ x+1 $, and the denominator is $ x^2 - 5x + 6 $. We can factor the denominator as $ (x-2)(x-3) $.

f(x) = \frac{x+1}{(x-2)(x-3)}

Evaluating f(x-1)

To evaluate $ f(x-1) $, we need to substitute $ x-1 $ into the function in place of $ x $. This means we will replace every instance of $ x $ with $ x-1 $.

f(x-1) = \frac{(x-1)+1}{(x-1)^2 - 5(x-1) + 6}

Simplifying the Expression

Now that we have substituted $ x-1 $ into the function, we can simplify the expression. Let's start by simplifying the numerator.

(x-1)+1 = x

So, the numerator simplifies to $ x $. Now, let's simplify the denominator.

(x-1)^2 - 5(x-1) + 6
= (x^2 - 2x + 1) - 5x + 5 + 6
= x^2 - 7x + 12

Factoring the Denominator

We can factor the denominator as $ (x-3)(x-4) $.

f(x-1) = \frac{x}{(x-3)(x-4)}

Conclusion

In this article, we evaluated the function $ f(x) = \frac{x+1}{x^2 - 5x + 6} $ at $ x-1 $. We substituted $ x-1 $ into the function in place of $ x $ and simplified the resulting expression. The final result is $ f(x-1) = \frac{x}{(x-3)(x-4)} $. This process demonstrates how to manipulate and simplify algebraic expressions.

Key Takeaways

  • To evaluate a function at a specific value, substitute that value into the function in place of the variable.
  • Simplify the resulting expression by combining like terms and factoring out common factors.
  • The final result should be a simplified expression that represents the value of the function at the given input.

Real-World Applications

Evaluating functions at specific values has numerous real-world applications. For example, in physics, we use functions to model the motion of objects. By evaluating these functions at specific times or positions, we can determine the object's velocity, acceleration, or position.

In economics, we use functions to model the behavior of markets or economies. By evaluating these functions at specific points in time, we can predict future trends or outcomes.

Common Mistakes

When evaluating functions, it's essential to be careful with the order of operations. Make sure to follow the correct order of operations (PEMDAS) to avoid errors.

Additionally, be mindful of the domain of the function. If the function is not defined at a specific value, the evaluation will result in an undefined or imaginary value.

Practice Problems

  1. Evaluate the function $ f(x) = \frac{x^2 - 4}{x + 2} $ at $ x-2 $.
  2. Simplify the expression $ \frac{(x-1)^2 - 5(x-1) + 6}{(x-1)+1} $.
  3. Evaluate the function $ f(x) = \frac{x^3 - 6x^2 + 11x - 6}{x^2 - 4x + 3} $ at $ x+1 $.

Conclusion

Introduction

In our previous article, we explored the concept of evaluating functions at specific values. We discussed how to substitute values into functions and simplify the resulting expressions. In this article, we will provide a Q&A guide to help you better understand and apply this concept.

Q: What is the purpose of evaluating functions?

A: Evaluating functions is a crucial concept in mathematics that helps us understand the behavior of functions at specific values. It has numerous real-world applications, such as modeling the motion of objects in physics, predicting future trends in economics, and optimizing systems in engineering.

Q: How do I evaluate a function at a specific value?

A: To evaluate a function at a specific value, substitute that value into the function in place of the variable. For example, to evaluate the function $ f(x) = \frac{x+1}{x^2 - 5x + 6} $ at $ x-1 $, you would substitute $ x-1 $ into the function in place of $ x $.

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

A: When evaluating functions, it's essential to be careful with the order of operations. Make sure to follow the correct order of operations (PEMDAS) to avoid errors. Additionally, be mindful of the domain of the function. If the function is not defined at a specific value, the evaluation will result in an undefined or imaginary value.

Q: How do I simplify the resulting expression after evaluating a function?

A: To simplify the resulting expression, combine like terms and factor out common factors. For example, if you evaluate the function $ f(x) = \frac{x^2 - 4}{x + 2} $ at $ x-2 $, you would simplify the resulting expression as follows:

f(x-2) = \frac{(x-2)^2 - 4}{(x-2)+2}
= \frac{x^2 - 4x + 4 - 4}{x}
= \frac{x^2 - 4x}{x}
= x - 4

Q: What are some real-world applications of evaluating functions?

A: Evaluating functions has numerous real-world applications, such as:

  • Modeling the motion of objects in physics
  • Predicting future trends in economics
  • Optimizing systems in engineering
  • Analyzing data in statistics
  • Solving problems in computer science

Q: How do I determine the domain of a function?

A: To determine the domain of a function, identify the values of the variable that make the function undefined or imaginary. For example, the function $ f(x) = \frac{1}{x} $ is undefined at $ x=0 $, so the domain of this function is all real numbers except $ x=0 $.

Q: What are some common functions that are used in real-world applications?

A: Some common functions that are used in real-world applications include:

  • Linear functions: $ f(x) = mx + b $
  • Quadratic functions: $ f(x) = ax^2 + bx + c $
  • Exponential functions: $ f(x) = a^x $
  • Logarithmic functions: $ f(x) = \log_a x $
  • Trigonometric functions: $ f(x) = \sin x, \cos x, \tan x $

Conclusion

Evaluating functions is a fundamental concept in mathematics that has numerous real-world applications. By understanding how to substitute values into functions and simplify the resulting expressions, you can apply this knowledge to solve problems in physics, economics, engineering, and other fields. Remember to be careful with the order of operations and the domain of the function to avoid errors. With practice, you'll become proficient in evaluating functions and simplifying expressions.

Practice Problems

  1. Evaluate the function $ f(x) = \frac{x^2 - 4}{x + 2} $ at $ x-2 $.
  2. Simplify the expression $ \frac{(x-1)^2 - 5(x-1) + 6}{(x-1)+1} $.
  3. Evaluate the function $ f(x) = \frac{x^3 - 6x^2 + 11x - 6}{x^2 - 4x + 3} $ at $ x+1 $.

Additional Resources

  • Khan Academy: Evaluating Functions
  • Mathway: Evaluating Functions
  • Wolfram Alpha: Evaluating Functions