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. $2x^2 - 16x + 15$
Introduction
In mathematics, function composition is a fundamental concept that allows us to combine two or more functions to create a new function. This concept is crucial in various areas of mathematics, including algebra, calculus, and analysis. In this article, we will explore the concept of function composition and provide a step-by-step guide on how to evaluate the composition of two functions.
What is Function Composition?
Function composition is the process 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, denoted by (f ∘ g)(x), is defined as:
(f ∘ g)(x) = f(g(x))
In other words, we first evaluate the function g(x) and then plug the result into the function f(x).
Evaluating Function Composition
To evaluate the composition of two functions, we need to follow a step-by-step approach. Let's consider the given functions:
p(x) = 2x^2 - 4x q(x) = x - 3
We want to find the composition (p ∘ q)(x). To do this, we need to follow these steps:
- Evaluate the inner function: First, we need to evaluate the inner function q(x). This means we need to plug in the value of x into the function q(x).
- Plug the result into the outer function: Once we have the result of the inner function, we need to plug it into the outer function p(x).
- Simplify the expression: Finally, we need to simplify the resulting expression to get the final answer.
Step-by-Step Evaluation
Let's follow the steps outlined above to evaluate the composition (p ∘ q)(x).
Step 1: Evaluate the Inner Function
First, we need to evaluate the inner function q(x). We plug in the value of x into the function q(x):
q(x) = x - 3
Step 2: Plug the Result into the Outer Function
Once we have the result of the inner function, we need to plug it into the outer function p(x):
p(q(x)) = 2(q(x))^2 - 4(q(x))
Step 3: Simplify the Expression
Now, we need to simplify the resulting expression:
p(q(x)) = 2(x - 3)^2 - 4(x - 3)
To simplify this expression, we need to expand the squared term:
p(q(x)) = 2(x^2 - 6x + 9) - 4(x - 3)
Next, we need to distribute the 2 and -4 to the terms inside the parentheses:
p(q(x)) = 2x^2 - 12x + 18 - 4x + 12
Finally, we need to combine like terms:
p(q(x)) = 2x^2 - 16x + 30
Conclusion
In this article, we explored the concept of function composition and provided a step-by-step guide on how to evaluate the composition of two functions. We used the given functions p(x) = 2x^2 - 4x and q(x) = x - 3 to evaluate the composition (p ∘ q)(x). By following the steps outlined above, we arrived at the final answer:
(p ∘ q)(x) = 2x^2 - 16x + 30
This result shows that the composition of the two functions is a new function that is a combination of the two original functions.
Key Takeaways
- Function composition is a fundamental concept in mathematics that allows us to combine two or more functions to create a new function.
- To evaluate the composition of two functions, we need to follow a step-by-step approach: evaluate the inner function, plug the result into the outer function, and simplify the expression.
- The composition of two functions is a new function that is a combination of the two original functions.
Practice Problems
To reinforce your understanding of function composition, try evaluating the following compositions:
- (f ∘ g)(x) where f(x) = x^2 + 1 and g(x) = 2x - 1
- (h ∘ k)(x) where h(x) = x^3 - 2 and k(x) = x + 1
Introduction
In our previous article, we explored the concept of function composition and provided a step-by-step guide on how to evaluate the composition of two functions. In this article, we will answer some frequently asked questions about function composition to help you better understand this concept.
Q: What is function composition?
A: Function composition is the process 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, denoted by (f ∘ g)(x), is defined as:
(f ∘ g)(x) = f(g(x))
Q: How do I evaluate the composition of two functions?
A: To evaluate the composition of two functions, you need to follow a step-by-step approach:
- Evaluate the inner function: Plug in the value of x into the inner function.
- Plug the result into the outer function: Take the result of the inner function and plug it into the outer function.
- Simplify the expression: Simplify the resulting expression to get the final answer.
Q: What is the difference between function composition and function evaluation?
A: Function composition is the process of combining two or more functions to create a new function, while function evaluation is the process of finding the value of a function at a given point. In other words, function composition is about creating a new function, while function evaluation is about finding the value of a function.
Q: Can I compose more than two functions?
A: Yes, you can compose more than two functions. For example, if you have three functions f(x), g(x), and h(x), you can compose them as follows:
(f ∘ g ∘ h)(x) = f(g(h(x)))
Q: What are some common applications of function composition?
A: Function composition has many applications in mathematics, science, and engineering. Some common applications include:
- Modeling real-world phenomena: Function composition can be used to model complex real-world phenomena, such as population growth or chemical reactions.
- Solving equations: Function composition can be used to solve equations, such as quadratic equations or differential equations.
- Analyzing data: Function composition can be used to analyze data, such as finding the average value of a function or the maximum value of a function.
Q: How do I know if a function is composite or not?
A: A function is composite if it can be expressed as the composition of two or more functions. In other words, if a function can be written as f(g(x)), where f and g are functions, then the function is composite.
Q: Can I use function composition to solve optimization problems?
A: Yes, function composition can be used to solve optimization problems. For example, if you want to maximize or minimize a function, you can use function composition to create a new function that represents the optimization problem.
Conclusion
In this article, we answered some frequently asked questions about function composition to help you better understand this concept. We hope that this article has been helpful in clarifying any doubts you may have had about function composition.
Key Takeaways
- Function composition is the process of combining two or more functions to create a new function.
- To evaluate the composition of two functions, you need to follow a step-by-step approach.
- Function composition has many applications in mathematics, science, and engineering.
- A function is composite if it can be expressed as the composition of two or more functions.
Practice Problems
To reinforce your understanding of function composition, try evaluating the following compositions:
- (f ∘ g)(x) where f(x) = x^2 + 1 and g(x) = 2x - 1
- (h ∘ k)(x) where h(x) = x^3 - 2 and k(x) = x + 1
- (f ∘ g ∘ h)(x) where f(x) = x^2 + 1, g(x) = 2x - 1, and h(x) = x + 1
Remember to follow the steps outlined above to evaluate the composition of the two functions.