If $p(x) = 2x^2 - 4x$ And $q(x) = X - 3$, What Is $(p \circ Q)(x$\]?A. $2x^2 - 4x + 12$ B. $2x^2 - 16x + 18$ C. $2x^2 - 16x + 30$ D. $2x^2 - 16x + 15$
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Introduction
In mathematics, a composition of functions is a way of combining two or more functions to create a new function. This is a fundamental concept in algebra and calculus, and it has numerous applications in various fields, including physics, engineering, and economics. In this article, we will explore the concept of composition of functions and provide a step-by-step guide on how to evaluate it.
What is Composition of Functions?
Composition of functions is a way of combining two or more functions to create a new function. Given two functions, f(x) and g(x), the composition of f and g is denoted by (f ∘ g)(x) 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.
Example: Composition of p(x) and q(x)
Let's consider two functions:
p(x) = 2x^2 - 4x q(x) = x - 3
We want to find the composition of p and q, denoted by (p ∘ q)(x). To do this, we need to substitute q(x) into p(x) in place of x.
Step 1: Substitute q(x) into p(x)
Substituting q(x) = x - 3 into p(x) = 2x^2 - 4x, we get:
(p ∘ q)(x) = p(q(x)) = 2(q(x))^2 - 4(q(x))
Step 2: Simplify the Expression
Now, we need to simplify the expression by expanding the square and combining like terms.
(p ∘ q)(x) = 2(q(x))^2 - 4(q(x)) = 2(x - 3)^2 - 4(x - 3) = 2(x^2 - 6x + 9) - 4x + 12 = 2x^2 - 12x + 18 - 4x + 12 = 2x^2 - 16x + 30
Conclusion
In this article, we have explored the concept of composition of functions and provided a step-by-step guide on how to evaluate it. We have also applied this concept to a specific example, where we found the composition of p(x) = 2x^2 - 4x and q(x) = x - 3. The final answer is (p ∘ q)(x) = 2x^2 - 16x + 30.
Answer Key
The correct answer is C. .
Tips and Tricks
- When evaluating the composition of functions, make sure to substitute the inner function into the outer function in place of x.
- Simplify the expression by expanding the square and combining like terms.
- Use the correct order of operations to evaluate the expression.
Practice Problems
- Find the composition of f(x) = x^2 + 2x and g(x) = x - 1.
- Find the composition of h(x) = 2x^2 - 3x and k(x) = x + 2.
- Find the composition of m(x) = x^2 - 4x and n(x) = x + 1.
Conclusion
In conclusion, composition of functions is a powerful tool in mathematics that allows us to combine two or more functions to create a new function. By following the steps outlined in this article, you can evaluate the composition of functions and apply it to various problems. Remember to substitute the inner function into the outer function in place of x, simplify the expression, and use the correct order of operations. With practice, you will become proficient in evaluating the composition of functions and applying it to real-world problems.
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Introduction
In our previous article, we explored the concept of composition of functions and provided a step-by-step guide on how to evaluate it. In this article, we will answer some frequently asked questions about composition of functions.
Q: What is the difference between composition of functions and function evaluation?
A: Composition of functions is a way of combining two or more functions to create a new function, whereas function evaluation is the process of finding the value of a function at a given input. In other words, composition of functions is a way of creating a new function, whereas function evaluation is a way of finding the value of a function.
Q: How do I know which function to substitute into the other?
A: When evaluating the composition of functions, you need to substitute the inner function into the outer function in place of x. In other words, you need to 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 so, as it can lead to infinite loops or other problems.
Q: How do I simplify the expression after substituting the inner function into the outer function?
A: After substituting the inner function into the outer function, you need to simplify the expression by expanding the square and combining like terms. This will give you the final expression for the composition of functions.
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 equation that can be solved using algebraic techniques.
Q: Are there any limitations to composition of functions?
A: Yes, there are limitations to composition of functions. For example, you cannot compose a function with itself if it is not invertible. Additionally, composition of functions can lead to infinite loops or other problems if not done carefully.
Q: Can I use composition of functions to model real-world problems?
A: Yes, you can use composition of functions to model real-world problems. By combining two or more functions, you can create a new function that models a complex system or process.
Q: How do I know if a composition of functions is invertible?
A: A composition of functions is invertible if and only if the inner function is invertible and the outer function is invertible. In other words, if the inner function has an inverse, and the outer function has an inverse, then the composition of functions is invertible.
Q: Can I use composition of functions to solve optimization problems?
A: Yes, you can use composition of functions to solve optimization problems. By combining two or more functions, you can create a new function that models a complex system or process, and then use optimization techniques to find the maximum or minimum value of the function.
Conclusion
In conclusion, composition of functions is a powerful tool in mathematics that allows us to combine two or more functions to create a new function. By following the steps outlined in this article, you can answer frequently asked questions about composition of functions and apply it to various problems. Remember to substitute the inner function into the outer function in place of x, simplify the expression, and use the correct order of operations. With practice, you will become proficient in evaluating the composition of functions and applying it to real-world problems.
Practice Problems
- Find the composition of f(x) = x^2 + 2x and g(x) = x - 1.
- Find the composition of h(x) = 2x^2 - 3x and k(x) = x + 2.
- Find the composition of m(x) = x^2 - 4x and n(x) = x + 1.
Answer Key
- (f ∘ g)(x) = x^2 + 2x - 1
- (h ∘ k)(x) = 2x^2 - 3x + 2
- (m ∘ n)(x) = x^2 - 4x + 1
Tips and Tricks
- When evaluating the composition of functions, make sure to substitute the inner function into the outer function in place of x.
- Simplify the expression by expanding the square and combining like terms.
- Use the correct order of operations to evaluate the expression.
- Be careful when substituting a function into itself, as it can lead to infinite loops or other problems.
- Use composition of functions to model real-world problems and solve optimization problems.