Ensuring Well-Defined Compositions: Domain Restrictions In Function Composition – A Math Perspective

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

domain

When we compose functions, we must make sure that the output of the first function is part of the domain of the second function. In other words, the input of the second function should be a valid output from the first function. This ensures that the composition of the functions is well-defined and that we can obtain a meaningful output.

For example, suppose we have two functions f(x) = x^2 and g(x) = sqrt(x). To compose these functions, we need to ensure that the output of f(x) = x^2 is a valid input for g(x) = sqrt(x). Since the domain of g(x) is non-negative real numbers, we need to ensure that the output of f(x) is also a non-negative real number. Therefore, we should restrict the domain of f(x) to non-negative real numbers, i.e., f(x) >= 0.

Thus, the composition of these two functions would be (g o f)(x) = g(f(x)) = sqrt(x^2) = |x|. In this case, we have ensured that the composition is well-defined by restricting the domain of f(x) to non-negative real numbers.

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