Monday, December 17, 2012

Arithmetic Combinations of Functions

Just as two real numbers can be combined by the operations of addition, subtraction, multiplication, and division to form other real numbers, two functions can be combined to create new functions.


Finding the Sum of Two Functions

Equation for the Sum:


To find the Sum of two functions, just like real numbers, simply add the two functions together.

Example:

Find for the functions and . Then evaluate the sum when .






Finding the Difference of Two Functions

Equation for the Difference:


To find the Difference of two equations, just like real numbers, simply subtract the equations. (Make sure to pay attention to the order in which the equations are to be subtracted).

Evaluate for the functions  and  when .










Finding the Product of Two Functions

Equation for the Product:





To find the Product of two equations, just like real numbers, simply multiply the equations. (Pay careful attention to exponents).

Given the functions and  , find the product of  and  . Then evaluate the product when  .














Find the Quotient of Two Functions

Equation for the Quotient:


 ,

 ,



To find the Quotient of two equations, just like real numbers, divide the equations. (Make sure to pay attention to the order in which the equations are to be divided). 

ATTENTION: If asked to find the domain of the new equation, which is very likely, one must check for solutions that don't work. This is required because of the rule that the denominator cannot equal zero.
Find and  for the functions  and  . Then find the domains of  and .


The quotient of  and  is:










The quotient of and is:








The domain of is  {see original equation}
The domain of is  {see original equation}



 The domain of is 


    The reason that there is a bracket around the "0" is because can equal zero without the answer being undefined and "0" is the lowest number in our set of inputs

  The reason that there is a parenthesis around the "2" is because  must be less than "2" otherwise the answer will be undefined

 The domain of is 
 
 The reason that there is a parenthesis around the "0" is because if equalled "0", the answer would be undefined because the square-root of zero is zero

The reason that there is a bracket around the "2" is because can be less than or equal to "2"
It must stop at "2" because that is the highest number of the domains in the previous sections: 
The domain of is  {see original equation}
                                  The domain of is  {see original equation}







Eric Smith

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