To understand composite functions, we first need to understand what a function is. A function is a set of ordered pairs (x, y) such that each x is paired with a unique y. In other words, a function maps each input x to a unique output y. For example, the function f(x) = 2x + 1 maps each input x to an output that is twice the input plus one. So, for example, if we input x = 2, the output is f(2) = 2(2) + 1 = 5.
Now, suppose we have two functions, f(x) and g(x), and we want to form a composite function by applying one function to the output of the other. The composite function is denoted by (f o g)(x), which is read as “f composed with g of x”. The idea is that we first apply g to the input x, and then apply f to the output of g. So, (f o g)(x) = f(g(x)).
To see why this works, let’s look at an example. Suppose we have the functions f(x) = 2x and g(x) = x + 1. To form the composite function (f o g)(x), we first apply g to the input x, which gives us g(x) = x + 1. Then, we apply f to the output of g, which gives us f(g(x)) = f(x + 1) = 2(x + 1) = 2x + 2. So, (f o g)(x) = 2x + 2.
One important thing to note about composite functions is that the order in which we apply the functions matters. That is, in general, (f o g)(x) is not the same as (g o f)(x). To see why this is true, let’s look at another example. Suppose we have the same functions as before, f(x) = 2x and g(x) = x + 1. This time, we form the composite function (g o f)(x) by first applying f to the input x, which gives us f(x) = 2x. Then, we apply g to the output of f, which gives us g(f(x)) = g(2x) = 2x + 1. So, (g o f)(x) = 2x + 1. Notice that this is not the same as the composite function we found earlier, (f o g)(x) = 2x + 2.
Another important concept related to composite functions is the idea of inverse functions. An inverse function is a function that “undoes” the action of another function. That is, if f(x) is a function and g(x) is its inverse, then g(f(x)) = x for all x in the domain of f. In other words, applying f and then g “undoes” the action of f, so we end up back where we started.
One way to find the inverse of a function is to switch the roles of x and y and solve for y in terms of x. For example, suppose we have the function f(x) = 2x + 1. To find its inverse, we switch x and y and solve for y:
x = 2y + 1 x – 1 = 2y
Definition of Composite Functions:
A composite function is a function that results from the combination of two or more functions. It is a function that performs the operations of one function on the output of another function. The notation used for composite functions is (f ? g)(x) or f(g(x)), where f and g are two functions, and x is an input.
To evaluate a composite function, we first apply the function g to the input x, and then apply the function f to the output of g(x). For example, suppose we have two functions f(x) and g(x). If we want to find the composite function (f ? g)(x), we first apply the function g to the input x, which gives us g(x). We then apply the function f to the output of g(x), which gives us f(g(x)).
Properties of Composite Functions:
Associativity: The composition of functions is associative, which means that the order in which we compose the functions does not matter. For example, if we have three functions f(x), g(x), and h(x), then (f ? g) ? h = f ? (g ? h).
Identity: The identity function is a function that returns the same value as its input. The composition of the identity function with any function f(x) is f(x). In other words, f(x) ? id(x) = f(x) = id(x) ? f(x).
Invertibility: If two functions f(x) and g(x) are invertible, then their composition (f ? g)(x) is also invertible. The inverse of (f ? g)(x) is (g?¹ ? f?¹)(x).
Domain and Range: The domain and range of a composite function depend on the domains and ranges of the individual functions involved in the composition. The domain of (f ? g)(x) is the set of all x such that g(x) is in the domain of f(x). The range of (f ? g)(x) is the set of all y such that y = f(g(x)) for some x in the domain of g(x).
Not commutative: Composition of functions is not commutative, which means that the order in which we compose the functions matters. For example, (f ? g)(x) is not necessarily the same as (g ? f)(x).
Examples of Composite Functions:
- f(g(x)) = sin(cos(x)): Here, the inner function g(x) is cos(x), and the outer function f(x) is sin(x).
- h(g(x)) = e^(x^2): In this case, the inner function g(x) is x, and the outer function h(x) is e^(x^2).
- f(g(x)) = ln(1 + sin(x)): The inner function g(x) is sin(x), and the outer function f(x) is ln(1 + x).
- g(f(x)) = sqrt(1 – cos(x)): The inner function f(x) is cos(x), and the outer function g(x) is sqrt(1 – x).
- h(g(f(x))) = 1/(1 + e^(-x)): Here, the innermost function f(x) is x, the middle function g(x) is e^x, and the outer function h(x) is 1/(1 + x).
Quiz
- What is a composite function? A composite function is a function that results from combining two or more functions, where the output of one function becomes the input of the next function.
- How is a composite function written? A composite function is written as f(g(x)), which means that the output of g(x) is used as the input of f(x).
- What is the domain of a composite function? The domain of a composite function is the set of all values of x for which the function is defined.
- How do you evaluate a composite function? To evaluate a composite function, you need to substitute the input value into the innermost function first and then work your way out, using the output of each function as the input of the next.
- What is the range of a composite function? The range of a composite function is the set of all possible output values that can be obtained from the function.
- What is the identity function? The identity function is a function that returns the input value unchanged. It is written as f(x) = x.
- What is the composition of an identity function with another function? The composition of an identity function with another function results in the same function. For example, f(g(x)) = g(x) if f(x) = x.
- What is the inverse of a composite function? The inverse of a composite function is the composition of the inverses of the individual functions in reverse order. For example, if f(g(x)) = h(x), then g(f^-1(x)) = h^-1(x).
- What is the domain of the inverse of a composite function? The domain of the inverse of a composite function is the range of the original composite function.
- What is the difference between a composite function and a simple function? A composite function is a combination of two or more functions, while a simple function is a function that consists of a single equation or formula. A simple function can also be a part of a composite function.
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