Arithmetic Combinations of Functions

Arithmetic Combinations of Functions

Functions can be combined arithmetically by the sum, difference, product, or quotient.

Sum:                           

Difference:             

Product:                        

Quotient:                        

The domain of an arithmetic combination of functions f and g includes all real numbers that are in both of the domains of f and g. In the case above, all of the domains are all real numbers except for the quotient f(x)/g(x). The denominator cannot be zero when combining functions with a quotient. In this quotient, g(x) cannot equal zero. Since g(x) cannot equal zero, zero acts as a vertical asymptote when graphing the combined function.

Example: If and, you can form the sum, difference, product, and quotient of f and g.

Sum:

Difference:

Product:

Quotient:

The denominator cannot equal zero. In this case, the denominator is (x - 1). Set the denominator to zero and isolate x to find the vertical asymptote. 

Add one to both sides

One is the vertical asymptote