Finding the Indicated Function

Oct 13, 2025 by miodeya.com 19 views

Introduction

In mathematics, function composition is a fundamental concept that involves combining two or more functions to create a new function. This process is essential in various areas of mathematics, including algebra, calculus, and analysis. In this article, we will explore the composition of two given functions, $s(x)$ and $t(x)$ , and find the indicated function $(s \cdot t)(x)$ . We will also determine the domain of the resulting function in interval notation.

Given Functions

The two given functions are:

$s(x) = \frac{x-2}{x^2-25}$

$t(x) = \frac{x-5}{2-x}$

Finding the Indicated Function

To find the indicated function $(s \cdot t)(x)$ , we need to multiply the two functions together. This involves multiplying the numerators and denominators separately.

$(s \cdot t)(x) = s(x) \cdot t(x)$

$= \frac{x-2}{x^2-25} \cdot \frac{x-5}{2-x}$

$= \frac{(x-2)(x-5)}{(x^2-25)(2-x)}$

Simplifying the Indicated Function

To simplify the indicated function, we can start by factoring the numerator and denominator.

$(s \cdot t)(x) = \frac{(x-2)(x-5)}{(x^2-25)(2-x)}$

$= \frac{(x-2)(x-5)}{(x-5)(x+5)(2-x)}$

Now, we can cancel out the common factor $(x-5)$ in the numerator and denominator.

$(s \cdot t)(x) = \frac{x-2}{(x+5)(2-x)}$

Domain of the Indicated Function

To determine the domain of the indicated function, we need to consider the restrictions on the values of $x$ that make the function undefined. In this case, the function is undefined when the denominator is equal to zero.

$(x+5)(2-x) = 0$

Solving for $x$ , we get:

$x+5 = 0 \Rightarrow x = -5$

$2-x = 0 \Rightarrow x = 2$

Therefore, the domain of the indicated function is all real numbers except $x = -5$ and $x = 2$ .

Interval Notation

The domain of the indicated function can be represented in interval notation as:

$(-\infty, -5) \cup (-5, 2) \cup (2, \infty)$

Conclusion

In this article, we found the indicated function $(s \cdot t)(x)$ by multiplying the two given functions $s(x)$ and $t(x)$ together. We also determined the domain of the resulting function in interval notation. The domain is all real numbers except $x = -5$ and $x = 2$ , which can be represented in interval notation as $(-\infty, -5) \cup (-5, 2) \cup (2, \infty)$ .

Future Work

In the next part of this series, we will explore the composition of the indicated function $(s \cdot t)(x)$ with another function. We will also determine the domain of the resulting function in interval notation.

References

[1] "Function Composition" by Math Open Reference
[2] "Domain and Range" by Khan Academy

Glossary

Function composition: The process of combining two or more functions to create a new function.
Domain: The set of all possible input values for a function.
Interval notation: A way of representing a set of real numbers using intervals.
Function Composition: Q&A ==========================

Introduction

In our previous article, we explored the composition of two given functions, $s(x)$ and $t(x)$ , and found the indicated function $(s \cdot t)(x)$ . We also determined the domain of the resulting function in interval notation. In this article, we will answer some frequently asked questions (FAQs) related to function composition.

Q&A

Q1: What is function composition?

A1: Function composition is the process of combining two or more functions to create a new function. This involves applying one function to the output of another function.

Q2: Why is function composition important?

A2: Function composition is important because it allows us to create new functions from existing ones. This is useful in various areas of mathematics, including algebra, calculus, and analysis.

Q3: How do I find the domain of a function?

A3: To find the domain of a function, you need to consider the restrictions on the values of $x$ that make the function undefined. This typically involves finding the values of $x$ that make the denominator equal to zero.

Q4: What is interval notation?

A4: Interval notation is a way of representing a set of real numbers using intervals. It is commonly used to represent the domain of a function.

Q5: How do I determine the domain of a composite function?

A5: To determine the domain of a composite function, you need to consider the restrictions on the values of $x$ that make the function undefined. This typically involves finding the values of $x$ that make the denominator equal to zero.

Q6: Can I use function composition to simplify a function?

A6: Yes, function composition can be used to simplify a function. By combining two or more functions, you can create a new function that is easier to work with.

Q7: How do I know if a function is composite or not?

A7: A function is composite if it can be expressed as the composition of two or more functions. This can be determined by analyzing the function and identifying any patterns or relationships between the input and output values.

Q8: Can I use function composition to solve equations?

A8: Yes, function composition can be used to solve equations. By applying a function to both sides of an equation, you can create a new equation that is easier to solve.

Q9: How do I determine the range of a composite function?

A9: To determine the range of a composite function, you need to consider the output values of the function. This typically involves analyzing the function and identifying any patterns or relationships between the input and output values.

Q10: Can I use function composition to model real-world phenomena?

A10: Yes, function composition can be used to model real-world phenomena. By combining two or more functions, you can create a new function that accurately represents a complex system or process.

Conclusion

In this article, we answered some frequently asked questions (FAQs) related to function composition. We hope that this article has provided you with a better understanding of function composition and its applications.

Future Work

In our next article, we will explore the composition of functions with multiple variables. We will also discuss the applications of function composition in various areas of mathematics and science.

References

[1] "Function Composition" by Math Open Reference
[2] "Domain and Range" by Khan Academy
[3] "Function Composition" by Wolfram MathWorld

Glossary

Function composition: The process of combining two or more functions to create a new function.
Domain: The set of all possible input values for a function.
Interval notation: A way of representing a set of real numbers using intervals.
Composite function: A function that is expressed as the composition of two or more functions.
Range: The set of all possible output values for a function.