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.
To put it simply, if we have two functions f and g, we can compose them to create a new function by feeding the output of f as the input of g. However, we need to ensure that the output of f is a valid input for g by checking that it lies within the domain of g.
For example, let’s say we have two functions f(x) = x^2 and g(x) = sqrt(x). If we want to compose these functions (g(f(x))), we need to ensure that the output of f(x) (which is x^2) is a valid input for g(x). Since the domain of g(x) is all non-negative real numbers, the output of f(x) (which can be any real number) needs to be limited to non-negative values by restricting the domain of f(x) to x >= 0. Thus, we can compose the functions f(x) and g(x) by writing g(f(x)) = g(x^2) = sqrt(x^2) = |x| (where |x| represents the absolute value of x).
So in summary, when composing functions, it is important to ensure that the first function’s output lies within the domain of the second function.
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