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When evaluating a composite function where we have either created or been given formulas, the rule of working from the inside out remains the same. The input value to the outer function will be the output of the inner function, which may be a numerical value, a variable name, or a more complicated expression.
While we can compose the functions for each individual input value, it is sometimes helpful to find a single formula that will calculate the result of a composition To do this, we will extend our idea of function evaluation. Recall that, when we evaluate a function like we substitute the value inside the parentheses into the formula wherever we see the input variable.
Given a formula for a composite function, evaluate the function.
Given and evaluate
Because the inside expression is we start by evaluating at 1.
Then so we evaluate at an input of 5.
As we discussed previously, the domain of a composite function such as is dependent on the domain of and the domain of It is important to know when we can apply a composite function and when we cannot, that is, to know the domain of a function such as Let us assume we know the domains of the functions and separately. If we write the composite function for an input as we can see right away that must be a member of the domain of in order for the expression to be meaningful, because otherwise we cannot complete the inner function evaluation. However, we also see that must be a member of the domain of otherwise the second function evaluation in cannot be completed, and the expression is still undefined. Thus the domain of consists of only those inputs in the domain of that produce outputs from belonging to the domain of Note that the domain of composed with is the set of all such that is in the domain of and is in the domain of
The domain of a composite function is the set of those inputs in the domain of for which is in the domain of
Given a function composition determine its domain.
Find the domain of
The domain of consists of all real numbers except since that input value would cause us to divide by 0. Likewise, the domain of consists of all real numbers except 1. So we need to exclude from the domain of that value of for which
So the domain of is the set of all real numbers except and This means that
We can write this in interval notation as
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