Select The Correct Answer.Consider These Functions:${ \begin{align*} f(x) &= X + 1 \ g(x) &= \frac{2}{x} \end{align*} }$Which Polynomial Is Equivalent To { (f \circ G)(x)$}$?A. { \frac{2}{x+1}$} B . \[ B. \[ B . \[ \frac{2x +
Introduction
In mathematics, composition of functions is a fundamental concept that allows us to combine two or more functions to create a new function. In this article, we will explore the composition of functions and use it to find the equivalent polynomial of a given composite function.
What is Composition of Functions?
Composition of functions is a process of combining two or more functions to create a new function. It is denoted by the symbol ∘ (circle) and is read as "composition of". For example, if we have two functions f(x) and g(x), then the composition of f and g is denoted by (f ∘ g)(x).
Step 1: Understanding the Given Functions
We are given two functions:
- f(x) = x + 1
- g(x) = 2/x
Step 2: Finding the Composition of Functions
To find the composition of functions, we need to substitute the output of one function into the other function. In this case, we will substitute g(x) into f(x).
(f ∘ g)(x) = f(g(x)) = f(2/x) = (2/x) + 1
Step 3: Simplifying the Expression
To simplify the expression, we can combine the terms:
(2/x) + 1 = 2/x + 1/x = (2 + 1)/x = 3/x
Conclusion
Therefore, the polynomial equivalent to (f ∘ g)(x) is 3/x.
Answer
The correct answer is:
C. 3/x
Discussion
Composition of functions is a powerful tool in mathematics that allows us to create new functions from existing ones. It is used extensively in various fields such as calculus, algebra, and engineering.
Example Use Cases
- Calculus: Composition of functions is used to find the derivative of a function. For example, if we have a function f(x) = x^2 and we want to find its derivative, we can use the composition of functions to find f'(x).
- Algebra: Composition of functions is used to solve equations and inequalities. For example, if we have an equation f(x) = g(x), we can use the composition of functions to find the solution.
- Engineering: Composition of functions is used to model real-world problems. For example, if we want to model the motion of an object, we can use the composition of functions to create a mathematical model.
Tips and Tricks
- Understand the given functions: Before finding the composition of functions, make sure you understand the given functions and their domains.
- Use substitution: To find the composition of functions, substitute the output of one function into the other function.
- Simplify the expression: After finding the composition of functions, simplify the expression to get the final answer.
Conclusion
Introduction
In our previous article, we explored the concept of composition of functions and used it to find the equivalent polynomial of a given composite function. In this article, we will answer some frequently asked questions about composition of functions.
Q&A
Q: What is the difference between composition of functions and function notation?
A: Composition of functions is a process of combining two or more functions to create a new function, whereas function notation is a way of writing a function as a mathematical expression. For example, if we have two functions f(x) and g(x), then the composition of f and g is denoted by (f ∘ g)(x), whereas the function notation for f(x) is simply f(x).
Q: How do I know if a function is composite or not?
A: A function is composite if it is the result of combining two or more functions. To determine if a function is composite, look for the symbol ∘ (circle) in the function notation. If you see the symbol ∘, then the function is composite.
Q: Can I compose functions with different domains?
A: Yes, you can compose functions with different domains. However, you need to make sure that the output of one function is within the domain of the other function. For example, if we have two functions f(x) = x + 1 and g(x) = 2/x, then we can compose them as (f ∘ g)(x) = f(g(x)) = (2/x) + 1.
Q: How do I simplify a composite function?
A: To simplify a composite function, you need to substitute the output of one function into the other function and then simplify the resulting expression. For example, if we have a composite function (f ∘ g)(x) = f(g(x)) = (2/x) + 1, then we can simplify it by combining the terms: (2/x) + 1 = 2/x + 1/x = (2 + 1)/x = 3/x.
Q: Can I compose functions with different variables?
A: Yes, you can compose functions with different variables. However, you need to make sure that the variables are compatible. For example, if we have two functions f(x) = x + 1 and g(y) = 2/y, then we can compose them as (f ∘ g)(x) = f(g(x)) = (2/x) + 1.
Q: How do I know if a composite function is equivalent to a given polynomial?
A: To determine if a composite function is equivalent to a given polynomial, you need to simplify the composite function and then compare it with the given polynomial. For example, if we have a composite function (f ∘ g)(x) = f(g(x)) = (2/x) + 1 and a given polynomial p(x) = 3/x, then we can simplify the composite function and compare it with the given polynomial: (2/x) + 1 = 2/x + 1/x = (2 + 1)/x = 3/x.
Tips and Tricks
- Understand the given functions: Before finding the composition of functions, make sure you understand the given functions and their domains.
- Use substitution: To find the composition of functions, substitute the output of one function into the other function.
- Simplify the expression: After finding the composition of functions, simplify the expression to get the final answer.
- Check the domains: Make sure that the output of one function is within the domain of the other function.
- Compare with the given polynomial: To determine if a composite function is equivalent to a given polynomial, simplify the composite function and compare it with the given polynomial.
Conclusion
In conclusion, composition of functions is a powerful tool in mathematics that allows us to create new functions from existing ones. By understanding the given functions, using substitution, and simplifying the expression, we can find the equivalent polynomial of a given composite function. We hope that this Q&A guide has helped you to better understand the concept of composition of functions.