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

Feb 27, 2025 by ADMIN 112 views

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Introduction

In mathematics, functions play a crucial role in representing relationships between variables. When evaluating a function, it is essential to understand the concept of function notation and how to apply it to different scenarios. In this article, we will explore the process of evaluating the function f(x-1) using the given function f(x) = (x+1)/(x^2 - 5x + 6).

Understanding the Function f(x)

The given function is f(x) = (x+1)/(x^2 - 5x + 6). 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) in the original function.

Evaluating f(x-1)

To evaluate f(x-1), we will substitute (x-1) into the function f(x) = (x+1)/(x^2 - 5x + 6). This gives us:

f(x-1) = ((x-1)+1)/((x-1)^2 - 5(x-1) + 6)

Simplifying the Expression

Now that we have substituted (x-1) into the function, we need to simplify the expression. Let's start by expanding the squared term:

((x-1)^2 - 5(x-1) + 6)

Using the formula (a-b)^2 = a^2 - 2ab + b^2, we can expand the squared term as follows:

(x-1)^2 = x^2 - 2x + 1

Now, substitute this expression back into the original equation:

f(x-1) = (x-1+1)/((x^2 - 2x + 1) - 5(x-1) + 6)

Further Simplification

Next, let's simplify the expression further by expanding the terms inside the parentheses:

(x^2 - 2x + 1) - 5(x-1) + 6

Using the distributive property, we can expand the terms as follows:

x^2 - 2x + 1 - 5x + 5 + 6

Combine like terms:

x^2 - 7x + 12

Now, substitute this expression back into the original equation:

f(x-1) = (x-1+1)/(x^2 - 7x + 12)

Final Simplification

Finally, let's simplify the expression by combining the terms in the numerator:

(x-1+1) = x

Now, substitute this expression back into the original equation:

f(x-1) = x/(x^2 - 7x + 12)

Conclusion

In this article, we evaluated the function f(x-1) using the given function f(x) = (x+1)/(x^2 - 5x + 6). We substituted (x-1) into the function, simplified the expression, and arrived at the final result: f(x-1) = x/(x^2 - 7x + 12). This demonstrates the importance of understanding function notation and how to apply it to different scenarios.

Discussion

What is the significance of evaluating functions like f(x-1)?
How does the process of evaluating functions like f(x-1) relate to real-world applications?
Can you think of other scenarios where evaluating functions like f(x-1) would be useful?

Additional Resources

For more information on functions and function notation, check out the following resources:
- Khan Academy: Functions
- Mathway: Function Notation
- Wolfram MathWorld: Function

Final Thoughts

Evaluating functions like f(x-1) is an essential skill in mathematics. By understanding function notation and how to apply it to different scenarios, we can solve a wide range of problems and explore new ideas. Whether you're a student, teacher, or simply someone interested in mathematics, I hope this article has provided you with a deeper understanding of evaluating functions like f(x-1).

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Introduction

In our previous article, we explored the process of evaluating the function f(x-1) using the given function f(x) = (x+1)/(x^2 - 5x + 6). We simplified the expression and arrived at the final result: f(x-1) = x/(x^2 - 7x + 12). In this article, we will answer some frequently asked questions related to evaluating functions like f(x-1).

Q&A

Q: What is the significance of evaluating functions like f(x-1)?

A: Evaluating functions like f(x-1) is essential in mathematics because it helps us understand the behavior of functions and how they change when we substitute different values. This is crucial in solving problems and making predictions in various fields, such as physics, engineering, and economics.

Q: How does the process of evaluating functions like f(x-1) relate to real-world applications?

A: The process of evaluating functions like f(x-1) has numerous real-world applications. For example, in physics, we use functions to describe the motion of objects, while in engineering, we use functions to design and optimize systems. In economics, we use functions to model the behavior of markets and make predictions about future trends.

Q: Can you think of other scenarios where evaluating functions like f(x-1) would be useful?

A: Yes, there are many scenarios where evaluating functions like f(x-1) would be useful. For example, in computer science, we use functions to write algorithms and solve problems, while in data analysis, we use functions to visualize and interpret data.

Q: What is the difference between f(x) and f(x-1)?

A: The main difference between f(x) and f(x-1) is the value of x. In f(x), x is the input value, while in f(x-1), x-1 is the input value. This means that f(x-1) is a shifted version of f(x), where the input value is decreased by 1.

Q: How do I know when to use f(x) and when to use f(x-1)?

A: You should use f(x) when the problem requires you to find the value of the function at a specific input value, while you should use f(x-1) when the problem requires you to find the value of the function at a shifted input value.

Q: Can I use f(x-1) to find the value of f(x)?

A: Yes, you can use f(x-1) to find the value of f(x) by substituting x+1 into the function f(x-1). This will give you the value of f(x).

Q: What is the relationship between f(x) and f(x-1)?

A: The relationship between f(x) and f(x-1) is that f(x-1) is a shifted version of f(x), where the input value is decreased by 1. This means that f(x-1) = f(x) - f'(x), where f'(x) is the derivative of f(x).

Conclusion

Additional Resources

For more information on functions and function notation, check out the following resources:
- Khan Academy: Functions
- Mathway: Function Notation
- Wolfram MathWorld: Function
For more information on evaluating functions like f(x-1), check out the following resources:
- MIT OpenCourseWare: Calculus
- Stanford University: Mathematics
- University of California, Berkeley: Mathematics