If $c(x)=4x-2$ And $d(x)=x^2+5x$, What Is $(c \circ D)(x$\]?A. $4x^3+18x^2-10x$ B. $x^2+9x-2$ C. $16x^2+4x-6$ D. $4x^2+20x-2$
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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 how to apply it to solve problems.
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 ∘ (circle). 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) 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 Two Functions
Let's consider two functions:
c(x) = 4x - 2 d(x) = x^2 + 5x
We want to find the composition of c and d, denoted by (c ∘ d)(x). To do this, we need to substitute the expression for d(x) into the expression for c(x).
Step 1: Substitute d(x) into c(x)
We substitute the expression for d(x) into the expression for c(x):
c(d(x)) = 4(d(x)) - 2 = 4(x^2 + 5x) - 2
Step 2: Simplify the Expression
We simplify the expression by distributing the 4 to the terms inside the parentheses:
c(d(x)) = 4x^2 + 20x - 2
Conclusion
Therefore, the composition of c and d is:
(c ∘ d)(x) = 4x^2 + 20x - 2
This is the final answer.
Comparison with Answer Choices
Let's compare our answer with the answer choices:
A. 4x^3 + 18x^2 - 10x B. x^2 + 9x - 2 C. 16x^2 + 4x - 6 D. 4x^2 + 20x - 2
Our answer matches answer choice D.
Tips and Tricks
When working with composition of functions, it's essential to follow the order of operations (PEMDAS):
- Parentheses: Evaluate expressions inside parentheses first.
- Exponents: Evaluate any exponential expressions next.
- Multiplication and Division: Evaluate any multiplication and division operations from left to right.
- Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.
By following this order of operations, you can ensure that you get the correct answer.
Practice Problems
Try solving the following practice problems:
- If f(x) = 2x + 1 and g(x) = x^2 - 3, find (f ∘ g)(x).
- If h(x) = x^2 + 2x and k(x) = 3x - 1, find (h ∘ k)(x).
Conclusion
In this article, we explored the concept of composition of functions and how to apply it to solve problems. We used the example of two functions c(x) and d(x) to demonstrate how to find the composition of c and d. We also compared our answer with the answer choices and provided tips and tricks for working with composition of functions. Finally, we provided practice problems for you to try.
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Introduction
In our previous article, we explored the concept of composition of functions and how to apply it to solve problems. In this article, we will provide a Q&A guide to help you better understand the concept of composition of functions.
Q: What is composition of functions?
A: Composition of functions is a way of combining two or more functions to create a new function. It is denoted by the symbol ∘ (circle). 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) and is defined as:
(f ∘ g)(x) = f(g(x))
Q: How do I find the composition of two functions?
A: To find the composition of two functions, you need to substitute the expression for one function into the expression for the other function. For example, if we have two functions c(x) and d(x), then the composition of c and d is:
(c ∘ d)(x) = c(d(x))
You need to substitute the expression for d(x) into the expression for c(x).
Q: What is the order of operations when working with composition of functions?
A: When working with composition of functions, it's essential to follow the order of operations (PEMDAS):
- Parentheses: Evaluate expressions inside parentheses first.
- Exponents: Evaluate any exponential expressions next.
- Multiplication and Division: Evaluate any multiplication and division operations from left to right.
- Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.
Q: Can I have multiple compositions of functions?
A: Yes, you can have multiple compositions of functions. For example, if we have three functions f(x), g(x), and h(x), then the composition of f, g, and h is:
(f ∘ g ∘ h)(x) = f(g(h(x)))
You need to substitute the expression for h(x) into the expression for g(x), and then substitute the expression for g(x) into the expression for f(x).
Q: How do I know which function to substitute first?
A: When working with multiple compositions of functions, it's essential to follow the order of operations. You need to substitute the innermost function first, and then work your way outwards.
Q: Can I use composition of functions to solve real-world problems?
A: Yes, composition of functions can be used to solve real-world problems. For example, in physics, you can use composition of functions to model the motion of an object. In economics, you can use composition of functions to model the behavior of a company.
Q: What are some common mistakes to avoid when working with composition of functions?
A: Some common mistakes to avoid when working with composition of functions include:
- Not following the order of operations
- Not substituting the innermost function first
- Not simplifying the expression after substitution
Conclusion
In this article, we provided a Q&A guide to help you better understand the concept of composition of functions. We covered topics such as the definition of composition of functions, how to find the composition of two functions, and common mistakes to avoid. We hope this guide has been helpful in your understanding of composition of functions.
Practice Problems
Try solving the following practice problems:
- If f(x) = 2x + 1 and g(x) = x^2 - 3, find (f ∘ g)(x).
- If h(x) = x^2 + 2x and k(x) = 3x - 1, find (h ∘ k)(x).
- If m(x) = x^3 + 2x and n(x) = x^2 - 1, find (m ∘ n)(x).
Resources
For more information on composition of functions, check out the following resources:
- Khan Academy: Composition of Functions
- Mathway: Composition of Functions
- Wolfram Alpha: Composition of Functions
We hope this guide has been helpful in your understanding of composition of functions. If you have any further questions, please don't hesitate to ask.