Given The Functions S ( X ) = X − 1 X 2 − 16 S(x)=\frac{x-1}{x^2-16} S ( X ) = X 2 − 16 X − 1 ​ And T ( X ) = X − 4 1 − X T(x)=\frac{x-4}{1-x} T ( X ) = 1 − X X − 4 ​ , Find The Expression For ( S + T ) ( X (s+t)(x ( S + T ) ( X ].

Introduction

In this article, we will explore the concept of function addition and how to find the expression for the sum of two given functions. We will be working with the functions $s(x)=\frac{x-1}{x^2-16}$ and $t(x)=\frac{x-4}{1-x}$. Our goal is to find the expression for $(s+t)(x)$, which represents the sum of the two functions.

Understanding the Functions

Before we proceed with finding the expression for $(s+t)(x)$, let's take a closer look at the two given functions.

Function s(x)

The function $s(x)$ is defined as $s(x)=\frac{x-1}{x^2-16}$. This function has a numerator of $x-1$ and a denominator of $x^2-16$. To simplify this function, we can factor the denominator as $(x+4)(x-4)$. Therefore, the function $s(x)$ can be rewritten as $s(x)=\frac{x-1}{(x+4)(x-4)}$.

Function t(x)

The function $t(x)$ is defined as $t(x)=\frac{x-4}{1-x}$. This function has a numerator of $x-4$ and a denominator of $1-x$. To simplify this function, we can rewrite the denominator as $-(x-1)$. Therefore, the function $t(x)$ can be rewritten as $t(x)=\frac{x-4}{-(x-1)}$.

Finding the Expression for (s+t)(x)

Now that we have a better understanding of the two functions, let's proceed with finding the expression for $(s+t)(x)$. To do this, we need to add the two functions together.

Adding the Functions

To add the two functions, we need to find a common denominator. In this case, the common denominator is $(x+4)(x-4)(x-1)$. Therefore, we can rewrite the functions as follows:

$s(x)=\frac{x-1}{(x+4)(x-4)}=\frac{(x-1)(x-1)}{(x+4)(x-4)(x-1)}$

$t(x)=\frac{x-4}{-(x-1)}=\frac{-(x-4)(x-1)}{(x+4)(x-4)(x-1)}$

Now that we have a common denominator, we can add the two functions together.

$(s+t)(x)=\frac{(x-1)(x-1)+(-)(x-4)(x-1)}{(x+4)(x-4)(x-1)}$

Simplifying the numerator, we get:

$(s+t)(x)=\frac{(x-1)^2-(x-4)(x-1)}{(x+4)(x-4)(x-1)}$

Expanding the numerator, we get:

$(s+t)(x)=\frac{x^2-2x+1-x^2+5x-4}{(x+4)(x-4)(x-1)}$

Simplifying the numerator, we get:

$(s+t)(x)=\frac{3x-3}{(x+4)(x-4)(x-1)}$

Conclusion

In this article, we found the expression for $(s+t)(x)$ by adding the two given functions together. We started by simplifying the functions and finding a common denominator. Then, we added the two functions together and simplified the resulting expression. The final expression for $(s+t)(x)$ is $\frac{3x-3}{(x+4)(x-4)(x-1)}$.

Final Answer

The final answer is $\boxed{\frac{3x-3}{(x+4)(x-4)(x-1)}}$.

Step-by-Step Solution

Here is the step-by-step solution to the problem:

1. Simplify the function $s(x)$ by factoring the denominator.
2. Simplify the function $t(x)$ by rewriting the denominator.
3. Find a common denominator for the two functions.
4. Rewrite the functions with the common denominator.
5. Add the two functions together.
6. Simplify the resulting expression.

Common Mistakes

Here are some common mistakes to avoid when solving this problem:

• Not simplifying the functions before adding them together.
• Not finding a common denominator before adding the functions together.
• Not simplifying the resulting expression after adding the functions together.

Real-World Applications

This problem has real-world applications in various fields, such as:

• Calculus: The concept of function addition is used in calculus to find the derivative of a function.
• Physics: The concept of function addition is used in physics to describe the motion of objects.
• Engineering: The concept of function addition is used in engineering to design and analyze systems.

Future Work

In the future, we can explore more complex problems involving function addition and subtraction. We can also explore the concept of function composition and how it relates to function addition and subtraction.

References

Here are some references that were used to solve this problem:

• [1] "Calculus" by Michael Spivak
• [2] "Physics for Scientists and Engineers" by Paul A. Tipler
• [3] "Engineering Mathematics" by John Bird

Note: The references provided are for illustrative purposes only and are not actual references used in the solution.

Introduction

In our previous article, we explored the concept of function addition and how to find the expression for the sum of two given functions. We worked with the functions $s(x)=\frac{x-1}{x^2-16}$ and $t(x)=\frac{x-4}{1-x}$ to find the expression for $(s+t)(x)$. In this article, we will answer some common questions that readers may have about the problem.

Q: What is the common denominator for the two functions?

A: The common denominator for the two functions is $(x+4)(x-4)(x-1)$.

Q: Why do we need to find a common denominator?

A: We need to find a common denominator to add the two functions together. Without a common denominator, we cannot add the functions.

Q: How do we simplify the numerator after adding the functions?

A: To simplify the numerator, we need to expand and combine like terms. In this case, we expanded the numerator and combined like terms to get $3x-3$.

Q: What is the final expression for $(s+t)(x)$?

A: The final expression for $(s+t)(x)$ is $\frac{3x-3}{(x+4)(x-4)(x-1)}$.

Q: Can we simplify the expression further?

A: Yes, we can simplify the expression further by factoring out a common factor from the numerator and denominator. However, in this case, the expression is already in its simplest form.

Q: What are some common mistakes to avoid when solving this problem?

A: Some common mistakes to avoid when solving this problem include not simplifying the functions before adding them together, not finding a common denominator before adding the functions together, and not simplifying the resulting expression after adding the functions together.

Q: What are some real-world applications of this problem?

A: This problem has real-world applications in various fields, such as calculus, physics, and engineering. The concept of function addition is used to describe the motion of objects, design and analyze systems, and find the derivative of a function.

Q: Can we use this problem to explore more complex concepts?

A: Yes, we can use this problem to explore more complex concepts, such as function composition and the properties of functions.

Q: What are some references that can be used to learn more about this topic?

A: Some references that can be used to learn more about this topic include "Calculus" by Michael Spivak, "Physics for Scientists and Engineers" by Paul A. Tipler, and "Engineering Mathematics" by John Bird.

Q: Can we use technology to solve this problem?

A: Yes, we can use technology, such as graphing calculators or computer algebra systems, to solve this problem. However, in this case, we solved the problem manually to illustrate the steps involved.

Q: What are some tips for solving this problem?

A: Some tips for solving this problem include simplifying the functions before adding them together, finding a common denominator before adding the functions together, and simplifying the resulting expression after adding the functions together.

Q: Can we use this problem to practice our skills in algebra and calculus?

A: Yes, we can use this problem to practice our skills in algebra and calculus. The problem involves simplifying expressions, finding common denominators, and adding functions together, all of which are important skills in algebra and calculus.

Q: What are some common misconceptions about this problem?

A: Some common misconceptions about this problem include thinking that we can add the functions together without finding a common denominator, thinking that the expression cannot be simplified further, and thinking that the problem is too difficult to solve.

Q: Can we use this problem to explore the concept of function composition?

A: Yes, we can use this problem to explore the concept of function composition. The problem involves adding two functions together, which is a key concept in function composition.

Q: What are some real-world examples of function composition?

A: Some real-world examples of function composition include the motion of objects, the design and analysis of systems, and the calculation of derivatives.

Q: Can we use this problem to practice our skills in problem-solving?

A: Yes, we can use this problem to practice our skills in problem-solving. The problem involves simplifying expressions, finding common denominators, and adding functions together, all of which are important skills in problem-solving.

Q: What are some tips for practicing problem-solving skills?

A: Some tips for practicing problem-solving skills include starting with simple problems, working through the steps involved, and practicing regularly.

Q: Can we use this problem to explore the concept of limits?

A: Yes, we can use this problem to explore the concept of limits. The problem involves adding two functions together, which is a key concept in limits.

Q: What are some real-world examples of limits?

A: Some real-world examples of limits include the calculation of derivatives, the design and analysis of systems, and the motion of objects.

Q: Can we use this problem to practice our skills in limits?

A: Yes, we can use this problem to practice our skills in limits. The problem involves adding two functions together, which is a key concept in limits.

Q: What are some tips for practicing limits skills?

A: Some tips for practicing limits skills include starting with simple problems, working through the steps involved, and practicing regularly.

Q: Can we use this problem to explore the concept of derivatives?

A: Yes, we can use this problem to explore the concept of derivatives. The problem involves adding two functions together, which is a key concept in derivatives.

Q: What are some real-world examples of derivatives?

A: Some real-world examples of derivatives include the calculation of rates of change, the design and analysis of systems, and the motion of objects.

Q: Can we use this problem to practice our skills in derivatives?

A: Yes, we can use this problem to practice our skills in derivatives. The problem involves adding two functions together, which is a key concept in derivatives.

Q: What are some tips for practicing derivatives skills?

A: Some tips for practicing derivatives skills include starting with simple problems, working through the steps involved, and practicing regularly.

Conclusion

In this article, we answered some common questions that readers may have about the problem of finding the expression for $(s+t)(x)$. We covered topics such as the common denominator, simplifying the numerator, and real-world applications. We also provided tips and references for further learning.