2. If F(x) = X²-2x And G(x) = X+1, What Is (gof)(x)? A) X² C) (x-1)2 D) X²+2x E) X²+2x+1 9(960) Xx 2anlatarak Çözer Misiniz
2. If f(x) = x²-2x and g(x) = x+1, what is (gof)(x)?
Understanding the Composition of Functions
In mathematics, the composition of functions is a way of combining two or more functions to create a new function. This is denoted by the symbol (f ∘ g)(x) or (g ∘ f)(x), where f and g are the individual functions. In this problem, we are asked to find the composition of the functions f(x) = x²-2x and g(x) = x+1.
Step 1: Understanding the Functions
Before we can find the composition of the functions, we need to understand what each function does. The function f(x) = x²-2x takes an input x and returns the value of x squared minus 2x. The function g(x) = x+1 takes an input x and returns the value of x plus 1.
Step 2: Finding the Composition
To find the composition of the functions, we need to plug in the function g(x) into the function f(x). This means that we will replace the x in the function f(x) with the function g(x). So, we get:
(f ∘ g)(x) = f(g(x))
Now, we need to substitute the function g(x) into the function f(x). We get:
(f ∘ g)(x) = (x+1)²-2(x+1)
Step 3: Simplifying the Expression
Now that we have the composition of the functions, we need to simplify the expression. We can start by expanding the squared term:
(x+1)² = x²+2x+1
Now, we can substitute this back into the expression:
(f ∘ g)(x) = x²+2x+1-2(x+1)
Step 4: Combining Like Terms
Now, we can combine like terms in the expression. We get:
(f ∘ g)(x) = x²+2x+1-2x-2
Step 5: Simplifying the Expression
Now, we can simplify the expression by combining like terms. We get:
(f ∘ g)(x) = x²+1
Conclusion
Therefore, the composition of the functions f(x) = x²-2x and g(x) = x+1 is (f ∘ g)(x) = x²+1.
Answer
The correct answer is E) x²+2x+1. However, this is not the correct answer. The correct answer is x²+1.
Explanation
The reason why the correct answer is not listed is because the question is asking for the composition of the functions, not the simplified expression. The composition of the functions is (f ∘ g)(x) = x²+1, but the simplified expression is x²+1-2x-2, which is not listed as an answer choice.
Note
It's worth noting that the question is asking for the composition of the functions, not the simplified expression. The composition of the functions is (f ∘ g)(x) = x²+1, but the simplified expression is x²+1-2x-2, which is not listed as an answer choice.
References
- [1] "Composition of Functions" by Khan Academy
- [2] "Composition of Functions" by Math Open Reference
Table of Contents
- [1.1] Understanding the Composition of Functions
- [1.2] Step 1: Understanding the Functions
- [1.3] Step 2: Finding the Composition
- [1.4] Step 3: Simplifying the Expression
- [1.5] Step 4: Combining Like Terms
- [1.6] Step 5: Simplifying the Expression
- [1.7] Conclusion
- [1.8] Answer
- [1.9] Explanation
- [1.10] Note
- [1.11] References
- [1.12] Table of Contents
Q&A: Composition of Functions
Understanding the Composition of Functions
In the previous article, we discussed the composition of functions and how to find the composition of two functions. In this article, we will answer some frequently asked questions about the composition of functions.
Q: What is the composition of functions?
A: The composition of functions is a way of combining two or more functions to create a new function. This is denoted by the symbol (f ∘ g)(x) or (g ∘ f)(x), where f and g are the individual functions.
Q: How do I find the composition of two functions?
A: To find the composition of two functions, you need to plug in one function into the other. This means that you will replace the x in the first function with the second function. For example, if we have the functions f(x) = x²-2x and g(x) = x+1, we can find the composition of the functions by plugging in g(x) into f(x).
Q: What is the difference between (f ∘ g)(x) and (g ∘ f)(x)?
A: The difference between (f ∘ g)(x) and (g ∘ f)(x) is the order in which the functions are composed. If we have the functions f(x) and g(x), then (f ∘ g)(x) means that we plug in g(x) into f(x), while (g ∘ f)(x) means that we plug in f(x) into g(x).
Q: Can I have multiple functions in a composition?
A: Yes, you can have multiple functions in a composition. For example, if we have the functions f(x), g(x), and h(x), we can find the composition of the functions by plugging in h(x) into g(x), and then plugging in the result into f(x).
Q: How do I simplify a composition of functions?
A: To simplify a composition of functions, you need to follow the order of operations. This means that you need to evaluate the functions inside the parentheses first, and then simplify the expression.
Q: What are some common mistakes to avoid when working with compositions of functions?
A: Some common mistakes to avoid when working with compositions of functions include:
- Not following the order of operations
- Not simplifying the expression correctly
- Not using the correct notation for the composition of functions
Q: How do I use the composition of functions in real-world applications?
A: The composition of functions is used in many real-world applications, including:
- Physics: The composition of functions is used to describe the motion of objects in terms of position, velocity, and acceleration.
- Engineering: The composition of functions is used to design and analyze complex systems, such as electronic circuits and mechanical systems.
- Computer Science: The composition of functions is used to write efficient and effective algorithms for solving complex problems.
Conclusion
In this article, we have answered some frequently asked questions about the composition of functions. We have discussed how to find the composition of two functions, how to simplify a composition of functions, and how to use the composition of functions in real-world applications.
References
- [1] "Composition of Functions" by Khan Academy
- [2] "Composition of Functions" by Math Open Reference
Table of Contents
- [1.1] Q: What is the composition of functions?
- [1.2] Q: How do I find the composition of two functions?
- [1.3] Q: What is the difference between (f ∘ g)(x) and (g ∘ f)(x)?
- [1.4] Q: Can I have multiple functions in a composition?
- [1.5] Q: How do I simplify a composition of functions?
- [1.6] Q: What are some common mistakes to avoid when working with compositions of functions?
- [1.7] Q: How do I use the composition of functions in real-world applications?
- [1.8] Conclusion
- [1.9] References
- [1.10] Table of Contents