If $f(x)=\frac{x-1}{3}$ And $g(x)=3x+1$, What Is $ ( F ∘ G ) ( X ) (f \circ G)(x) ( F ∘ G ) ( X ) [/tex]?A. $(f \circ G)(x)=3x+1$ B. $(f \circ G)(x)=x-3$ C. $ ( F ∘ G ) ( X ) = 3 X (f \circ G)(x)=3x ( F ∘ G ) ( X ) = 3 X [/tex] D. $(f \circ
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Introduction
In mathematics, a function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. Composition of functions is a way of combining two or more functions to create a new function. In this article, we will explore the concept of composition of functions and use it to find the composition of two given functions, f(x) and g(x).
What is Composition of Functions?
Composition of functions is a way of combining two or more functions to create a new function. It is denoted by the symbol ∘ and is defined as:
(f ∘ g)(x) = f(g(x))
In other words, we first apply the function g to the input x, and then apply the function f to the result.
Given Functions
We are given two functions:
f(x) = (x-1)/3
g(x) = 3x + 1
Composition of Functions
To find the composition of these two functions, we need to substitute g(x) into f(x) in place of x.
(f ∘ g)(x) = f(g(x))
= f(3x + 1)
= ((3x + 1) - 1)/3
= (3x)/3
= x
Simplifying the Expression
We can simplify the expression further by canceling out the common factor of 3 in the numerator and denominator.
(f ∘ g)(x) = x
Conclusion
In this article, we have explored the concept of composition of functions and used it to find the composition of two given functions, f(x) and g(x). We have shown that the composition of these two functions is simply x.
Answer
The correct answer is:
C. (f ∘ g)(x) = 3x
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 calculus, algebra, and other branches of mathematics.
In this example, we have shown that the composition of two simple functions can result in a very simple function. However, in more complex cases, the composition of functions can lead to much more complex and interesting results.
Real-World Applications
Composition of functions has many real-world applications in fields such as physics, engineering, and economics. For example, in physics, the composition of functions can be used to model the motion of objects under the influence of gravity or other forces.
In engineering, composition of functions can be used to design and optimize complex systems, such as electronic circuits or mechanical systems.
In economics, composition of functions can be used to model the behavior of economic systems and make predictions about future trends.
Tips and Tricks
When working with composition of functions, it is essential to remember the following tips and tricks:
- Always substitute the inner function into the outer function in place of x.
- Simplify the expression as much as possible by canceling out common factors.
- Use the correct notation for composition of functions, which is ∘.
By following these tips and tricks, you can master the concept of composition of functions and use it to solve a wide range of problems in mathematics and other fields.
Conclusion
In conclusion, composition of functions is a powerful tool in mathematics that allows us to create new functions from existing ones. It has many real-world applications in fields such as physics, engineering, and economics. By mastering the concept of composition of functions, you can solve a wide range of problems in mathematics and other fields.
Final Answer
The final answer is:
C. (f ∘ g)(x) = 3x
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Introduction
In our previous article, we explored the concept of composition of functions and used it to find the composition of two given functions, f(x) and g(x). In this article, we will answer some frequently asked questions about composition of functions.
Q: What is the difference between function composition and function addition?
A: Function composition and function addition are two different mathematical operations. Function composition involves combining two or more functions to create a new function, while function addition involves adding two or more functions together.
Q: How do I know which function to substitute into the other?
A: When substituting one function into another, you should substitute the inner function into the outer function in place of x. This means that you should replace x in the outer function with the expression for the inner function.
Q: Can I substitute a function into itself?
A: Yes, you can substitute a function into itself. This is known as a recursive function. However, be careful when doing this, as it can lead to infinite loops or other problems.
Q: How do I simplify a composition of functions?
A: To simplify a composition of functions, you should first substitute the inner function into the outer function. Then, you should simplify the resulting expression by canceling out common factors.
Q: Can I use composition of functions to solve equations?
A: Yes, you can use composition of functions to solve equations. By substituting one function into another, you can create a new function that can be used to solve the equation.
Q: What are some real-world applications of composition of functions?
A: Composition of functions has many real-world applications in fields such as physics, engineering, and economics. For example, in physics, composition of functions can be used to model the motion of objects under the influence of gravity or other forces. In engineering, composition of functions can be used to design and optimize complex systems, such as electronic circuits or mechanical systems.
Q: How do I know if a composition of functions is a one-to-one function?
A: A composition of functions is a one-to-one function if and only if the inner function is a one-to-one function. This means that the inner function must pass the horizontal line test, which means that no horizontal line intersects the graph of the function in more than one place.
Q: Can I use composition of functions to find the inverse of a function?
A: Yes, you can use composition of functions to find the inverse of a function. By substituting the inverse function into the original function, you can create a new function that is the inverse of the original function.
Conclusion
In conclusion, composition of functions is a powerful tool in mathematics that allows us to create new functions from existing ones. By mastering the concept of composition of functions, you can solve a wide range of problems in mathematics and other fields.
Final Answer
The final answer is:
- Composition of functions is a way of combining two or more functions to create a new function.
- To simplify a composition of functions, you should first substitute the inner function into the outer function, and then simplify the resulting expression by canceling out common factors.
- Composition of functions has many real-world applications in fields such as physics, engineering, and economics.
- A composition of functions is a one-to-one function if and only if the inner function is a one-to-one function.
- You can use composition of functions to find the inverse of a function by substituting the inverse function into the original function.
Additional Resources
If you want to learn more about composition of functions, we recommend the following resources:
- Khan Academy: Composition of Functions
- Mathway: Composition of Functions
- Wolfram Alpha: Composition of Functions
By following these resources and practicing the concept of composition of functions, you can master this powerful tool in mathematics and use it to solve a wide range of problems in mathematics and other fields.