If $f(x) = X^2 + 1$ And $g(x) = X - 4$, Which Value Is Equivalent To $ ( F ∘ G ) ( 10 ) (f \circ G)(10) ( F ∘ G ) ( 10 ) [/tex]?A. 37 B. 97 C. 126 D. 606
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 branches of mathematics, including algebra, calculus, and analysis. In this article, we will explore the concept of function composition and apply it to a specific problem involving two given functions, f(x) and g(x).
What is Function Composition?
Function composition is the process of combining two or more functions to create a new function. This is done by applying the output of one function as the input to another function. In other words, if we have two functions, f(x) and g(x), then the composition of f and g, denoted as (f ∘ g)(x), is defined as:
(f ∘ g)(x) = f(g(x))
Given Functions
In this problem, we are given two functions:
f(x) = x^2 + 1 g(x) = x - 4
The Problem
We are asked to find the value of (f ∘ g)(10), which means we need to find the value of f(g(10)).
Step 1: Evaluate g(10)
To find g(10), we substitute x = 10 into the function g(x) = x - 4.
g(10) = 10 - 4 g(10) = 6
Step 2: Evaluate f(g(10))
Now that we have the value of g(10), we can substitute it into the function f(x) = x^2 + 1.
f(g(10)) = f(6) f(6) = 6^2 + 1 f(6) = 36 + 1 f(6) = 37
Conclusion
Therefore, the value of (f ∘ g)(10) is equivalent to 37.
Why is Function Composition Important?
Function composition is an essential concept in mathematics because it allows us to create new functions from existing ones. This concept has numerous applications in various fields, including:
- Algebra: Function composition is used to solve equations and inequalities.
- Calculus: Function composition is used to find derivatives and integrals.
- Analysis: Function composition is used to study the properties of functions, such as continuity and differentiability.
Real-World Applications
Function composition has numerous real-world applications, including:
- Computer Science: Function composition is used in programming languages to create new functions from existing ones.
- Engineering: Function composition is used to model complex systems and solve problems in fields like physics and engineering.
- Economics: Function composition is used to model economic systems and make predictions about economic trends.
Conclusion
Introduction
In our previous article, we explored the concept of function composition and applied it to a specific problem involving two given functions, f(x) and g(x). 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. This is done by applying the output of one function as the input to another function.
Q: How do I evaluate a function composition?
A: To evaluate a function composition, you need to follow these steps:
- Evaluate the inner function (g(x)) at the given input (x).
- Substitute the value of the inner function into the outer function (f(x)).
- Evaluate the outer function using the substituted value.
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 input.
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), then the composition of f, g, and h is denoted as (f ∘ g ∘ h)(x) and is defined as:
(f ∘ g ∘ h)(x) = f(g(h(x)))
Q: How do I know which function to evaluate first in a function composition?
A: To determine which function to evaluate first, you need to look at the order of the functions in the composition. The innermost function is evaluated first, followed by the next function, and so on.
Q: Can I use function composition to solve equations and inequalities?
A: Yes, function composition can be used to solve equations and inequalities. By applying function composition to both sides of an equation or inequality, you can simplify the expression and solve for the unknown variable.
Q: What are some real-world applications of function composition?
A: Function composition has numerous real-world applications, including:
- Computer Science: Function composition is used in programming languages to create new functions from existing ones.
- Engineering: Function composition is used to model complex systems and solve problems in fields like physics and engineering.
- Economics: Function composition is used to model economic systems and make predictions about economic trends.
Q: How do I practice function composition?
A: To practice function composition, you can try the following:
- Start with simple functions and gradually move to more complex ones.
- Practice evaluating function compositions using different inputs and outputs.
- Try to solve problems involving function composition, such as solving equations and inequalities.
Conclusion
In conclusion, function composition is a powerful tool in mathematics that allows us to create new functions from existing ones. By understanding function composition, we can solve problems in various fields and make predictions about complex systems. We hope this Q&A article has helped you better understand function composition and its applications.