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, let’s first define what a function is. In mathematics, a function is a rule that assigns a unique output value to each input value. The set of all possible input values for a function is called its domain, and the set of all possible output values is called its range.
When we compose two functions, say f and g, we perform the operation g(f(x)), where we first apply the function f to an input value x, and then we apply the function g to the output of f.
However, in order for this composition to be valid, the output of f must be part of the domain of g. In other words, the value obtained by applying f to x must be an acceptable input for g. If this condition is not satisfied, the composition is not defined.
For example, let’s consider two functions f(x) = √x and g(x) = 1/x. The domain of f is all non-negative real numbers, since the square root is only defined for non-negative values. On the other hand, the domain of g is all real numbers except zero, since division by zero is undefined.
If we want to compose these functions by evaluating g(f(x)), we need to ensure that the output of f is not zero (because g is undefined at zero). Therefore, the domain of f must only include non-zero values, and thus, the domain of x for the composition would be all positive real numbers.
To summarize, when composing functions, we must ensure that the output of the first function is a valid input for the second function by making sure it belongs to the domain of the second function.
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