The Importance of Checking the Domain Before Composing Functions | Explained with Examples

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, we must make sure that the output of the first function is part of the domain of the second function

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 understand why this is important, let’s first review what it means to compose functions. When we compose two functions f and g, we create a new function, denoted as g(f(x)), by plugging the output of f into g. In other words, we take the output of f as the input for g.

For this composition to be valid, the output of f must be a legitimate input for g. This means that the output of f must be within the domain of g. The domain of a function represents all the possible values that the function can accept as inputs.

For example, let’s consider two functions f(x) = √x and g(x) = 2x + 1. The domain of f is all non-negative real numbers (x ≥ 0), as the square root function is valid only for non-negative values. The domain of g, on the other hand, is all real numbers.

If we want to compose these two functions, we must ensure that the output of f is within the domain of g. Since the output of f is always non-negative, it satisfies the domain requirement of g. Consequently, we can compose f and g by plugging f(x) into g, like this: g(f(x)) = 2√x + 1.

In general, to compose functions, we need to consider the domains of both functions involved. The output of the first function should be within the domain of the second function to ensure that the composition is valid.

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