Understanding Function Composition | Ensuring Proper Domain Compatibility

1) 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 this concept better, let’s start by defining what a domain is. In mathematics, the domain of a function refers to the set of all possible input values for which the function is defined. In other words, it represents the valid inputs that we can plug into a function.

When we compose functions, we take the output of one function and use it as the input for another function. However, it is important to note that not all values can be used as inputs for every function. Each function has its own specific domain, which defines the set of valid inputs.

So, when we compose functions, we need to ensure that the output of the first function falls within the domain of the second function. Otherwise, if the output of the first function is not a valid input for the second function, the composition will not be defined.

For example, let’s consider two functions: f(x) = x^2 and g(x) = 1/x. The domain of f(x) is all real numbers because squaring any real number will always give a valid result. However, the domain of g(x) is all real numbers except 0 because dividing by zero is undefined.

If we want to compose these functions by taking the output of f(x) and plugging it into g(x) (g(f(x))), we need to make sure that the output of f(x) does not include 0. Otherwise, we would be attempting to divide by zero, which is not allowed. Hence, the output of f(x) should be a real number other than 0 for the composition to be defined.

In summary, when we compose functions, the output of the first function must belong to the domain of the second function to ensure that the composition is well-defined.

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