Why the Domain of Functions is Crucial in Composing Functions

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 need to ensure that the output of the first function belongs to the domain of the second function

When we compose functions, we need to ensure that the output of the first function belongs to the domain of the second function.

The domain of a function consists of all possible input values or x-values for which the function is defined. In other words, it is the set of all the values that can be substituted into the function to produce valid output or y-values.

When we compose two functions, let’s call them f(x) and g(x), we generally write it as (g ∘ f)(x), which means we apply the function f(x) first, and then feed its output into the function g(x).

To ensure the output of the first function is part of the domain of the second function, we must make sure that the values obtained from applying the first function, f(x), are valid input values for the second function, g(x).

So, when composing functions, we need to consider the following:

1. Determine the domain of the first function, f(x).
2. Identify the output values obtained after applying f(x).
3. Check if these output values fall within the domain of the second function, g(x).

If the output values of f(x) do not lie within the domain of g(x), then the composition of the two functions is not possible as we cannot apply the second function to those particular values.

It is crucial to pay attention to the domains of functions while composing them to ensure that the resulting composition is valid and well-defined.

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