For Which Pair Of Functions Is { (g \circ F)(a)=|a|-2$}$?A. { F(a)=a^2-4$}$ And { G(a)=\sqrt{a}$}$B. { F(a)=\frac{1}{2} A-1$}$ And { G(a)=2 A-2$}$C. { F(a)=5+a^2$}$ And
Introduction
In mathematics, the composition of functions is a fundamental concept that plays a crucial role in various branches of mathematics, including algebra, calculus, and analysis. The 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 provide a step-by-step guide on how to determine the composition of two functions.
What is Composition of Functions?
The 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 (g ∘ f)(x) and is defined as:
(g ∘ f)(x) = g(f(x))
In other words, the composition of f and g is a new function that takes an input x, applies the function f to it, and then applies the function g to the result.
Example 1: Composition of Functions
Let's consider two functions f(x) = x^2 - 4 and g(x) = √x. We want to find the composition of f and g, denoted by (g ∘ f)(x).
(g ∘ f)(x) = g(f(x)) = g(x^2 - 4) = √(x^2 - 4)
Example 2: Composition of Functions
Let's consider two functions f(x) = (1/2)x - 1 and g(x) = 2x - 2. We want to find the composition of f and g, denoted by (g ∘ f)(x).
(g ∘ f)(x) = g(f(x)) = g((1/2)x - 1) = 2((1/2)x - 1) - 2 = x - 4
Example 3: Composition of Functions
Let's consider two functions f(x) = 5 + x^2 and g(x) = √x. We want to find the composition of f and g, denoted by (g ∘ f)(x).
(g ∘ f)(x) = g(f(x)) = g(5 + x^2) = √(5 + x^2)
Problem: Composition of Functions
For which pair of functions is (g ∘ f)(a) = |a| - 2?
A. f(a) = a^2 - 4 and g(a) = √a B. f(a) = (1/2)a - 1 and g(a) = 2a - 2 C. f(a) = 5 + a^2 and g(a) = √a
Solution
To determine the correct pair of functions, we need to find the composition of each pair and check if it satisfies the given equation (g ∘ f)(a) = |a| - 2.
Pair A: f(a) = a^2 - 4 and g(a) = √a
(g ∘ f)(a) = g(f(a)) = g(a^2 - 4) = √(a^2 - 4) = |a| - 2
Pair B: f(a) = (1/2)a - 1 and g(a) = 2a - 2
(g ∘ f)(a) = g(f(a)) = g((1/2)a - 1) = 2((1/2)a - 1) - 2 = a - 4
Pair C: f(a) = 5 + a^2 and g(a) = √a
(g ∘ f)(a) = g(f(a)) = g(5 + a^2) = √(5 + a^2) ≠|a| - 2
Conclusion
Based on the calculations above, we can see that only Pair A satisfies the given equation (g ∘ f)(a) = |a| - 2. Therefore, the correct pair of functions is:
A. f(a) = a^2 - 4 and g(a) = √a
Final Answer
Q&A: 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. Given two functions f(x) and g(x), the composition of f and g is denoted by (g ∘ f)(x) and is defined as:
(g ∘ f)(x) = g(f(x))
Q: How do I determine the composition of two functions?
A: To determine the composition of two functions, you need to apply the function f to the input x, and then apply the function g to the result. This can be done by substituting the expression for f(x) into the function g.
Q: What is the difference between the composition of functions and the product of functions?
A: The composition of functions is a way of combining two or more functions to create a new function, whereas the product of functions is a way of multiplying two or more functions together. For example, if we have two functions f(x) and g(x), the composition of f and g is denoted by (g ∘ f)(x), whereas the product of f and g is denoted by f(x)g(x).
Q: Can the composition of functions be commutative?
A: No, the composition of functions is not commutative. This means that the order of the functions matters, and the composition of f and g is not necessarily the same as the composition of g and f.
Q: What is the identity function in the context of composition of functions?
A: The identity function is a function that leaves its input unchanged. In the context of composition of functions, the identity function is denoted by I(x) = x. This means that if we compose the identity function with any other function f, the result is the same as the original function f.
Q: Can the composition of functions be used to solve equations?
A: Yes, the composition of functions can be used to solve equations. By applying the composition of functions to both sides of an equation, we can simplify the equation and solve for the unknown variable.
Q: What are some common applications of the composition of functions?
A: The composition of functions has many applications in mathematics, science, and engineering. Some common applications include:
- Modeling real-world phenomena, such as population growth or chemical reactions
- Solving equations and inequalities
- Finding the inverse of a function
- Analyzing the behavior of complex systems
Q: How do I determine the domain and range of a composite function?
A: To determine the domain and range of a composite function, you need to consider the domains and ranges of the individual functions involved in the composition. The domain of the composite function is the set of all possible input values, while the range is the set of all possible output values.
Q: Can the composition of functions be used to find the inverse of a function?
A: Yes, the composition of functions can be used to find the inverse of a function. By applying the composition of functions to both sides of an equation, we can solve for the inverse of the function.
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 considering the domains and ranges of the individual functions involved in the composition
- Not applying the composition of functions correctly
- Not checking for errors in the composition of functions
Conclusion
In conclusion, the composition of functions is a powerful tool that can be used to solve equations, find the inverse of a function, and analyze the behavior of complex systems. By understanding the basics of composition of functions, you can apply this concept to a wide range of problems and applications.