Evaluating Functions
Afunctionis a mapping between an input and an output. For example, the function takes an input and returns its square In this case, and so on. The key difference between a function and a more generalrelationis that for every input to a function, there is exactly one output.
What is the value of the function at ?
We see that we want to evaluate , where . Thus,
Sometimes when mapping between an input and output, the input can be another function that maps to another input. This is called acomposite function. When evaluating a composite function, first we compose the function and evaluate the result as we do any other function.
Given that and what is the value of the composite function at ?
First compose the function:
Now, evaluate the function at
Given that and , what is the value of the function at ?
First compose the function:
Now, evaluate at
Given that and , for what value of is ?
Compose the function and evaluate at
Equate this to to obtain
What is the value of the function at ?
We could just plug in in every place of but notice that when which will collapse all the product terms with it. Thus,
Sometimes a function is given as a piecewise defined function, which is a function defined by multiple sub-functions. Each sub-function is defined by a certain interval or conditions.
如果函数 is defined by then find the values, if exists, of
Note that the domain of is
Since for , we have
2.5 does not belong to domain , is not defined.
Since , we have
Since , we have
Since , we have
Since , we have
Given the following function: what is the value of ?
Based on the piecewised function above, if we evaluate over the function If is between and inclusive, then we evaluate over the function If we evaluate over the function