Understanding Function Composition in Mathematics | Ensuring Valid Inputs for Successful Compositions

When we compose functions, we must make sure that the output of the first function is part of the ___ of the second function.

When we compose functions, the output of the first function should be part of the domain of the second function

When we compose functions, the output of the first function should be part of the domain of the second function.

In mathematics, function composition involves taking the output of one function and using it as the input for another function. When composing functions f(x) and g(x), the output of f(x) is denoted as f(g(x)), which means that the output of g(x) is used as the input for f(x).

However, it is important to ensure that the output of the first function, g(x), is a valid input for the second function, f(x). This means that the output of g(x) must be part of the domain of f(x) so that the composition can be evaluated.

For example, let’s consider two functions: f(x) = √(x) and g(x) = x^2. To find f(g(x)), we first evaluate g(x) which gives us x^2. Then, we substitute this output into f(x), which gives us √(x^2). In this case, it is crucial to note that the output of g(x), which is x^2, must be non-negative since the square root function only accepts non-negative inputs. Therefore, the domain of g(x) must be such that x^2 is non-negative so that the composition f(g(x)) is defined.

In general, when composing functions, we need to consider the domain of each function and make sure that the outputs of the preceding functions are suitable inputs for the subsequent functions.

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