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Imagine you have two function machines, let's call them machine and machine . Each machine has its own rules, which are its function ( and ) and the raw materials it can process (its domain, and ). We can combine these two machines to create a new machine using addition or subtraction operations.
If we want to add function and function , we simply add the results from each function for the same value of . The result is a new function we call .
Important note: The combined machine can only process raw materials (values of ) that can be processed by both original machines, and . So, the domain (domain of origin) of the function is the intersection of the domain of and the domain of .
This means that must be a member of AND also a member of .
Suppose we have two functions:
Step : Determine the resulting function from addition
Step : Determine the domain of the resulting function
We find the intersection of and :
So, the resulting function from addition is with domain .
The process is similar to addition. To subtract function from function , we subtract the result of from for the same value of . The result is a new function .
Its domain is also the same as for addition, namely the intersection of the domain of and the domain of . Why? Because again, the value of must be processable by both initial functions before it can be subtracted.
We use the same functions as in the addition example:
Step : Determine the resulting function from subtraction
Step : Determine the domain of the resulting function Its domain is the same as the domain of the addition result because the intersection rule is the same:
So, the resulting function from subtraction is with domain .
Given the functions with and function with .
Finding :
Finding Domain :
So, with domain all real numbers.
Finding :
Calculating :
We use the result from number :
Calculating :
We use the result from number :
Finding Domain :
So, with domain all real numbers.
