A good portion of the math you will study in Precalculus deals with functions. If you are reading this article I’m assuming you have some basic knowledge about functions. Rather than trying to go over these basics, I’ll jump right into providing some definitions of the different kinds of operations you can do with functions to create new functions.
If two functions f and g are both defined at a real number x, and if f(x) and g(x) are both real numbers, then it is possible to perform real number operations such as addition, subtraction, multiplication, or division with f(x) and g(x) . Here these operations are defined in detail:
Sum function: (f+g)(x) = f(x) + g(x)
Difference function: (f-g)(x) = f(x) – g(x)
Product function: (fg)(x) = f(x)g(x)
Quotient function: (f/g)(x) = f(x)/g(x) where g(x) cannot equal 0
The domain of each of these functions is defined on the intersection of the domains of f and g , with the exception that the values of x where g(x) = 0 must be excluded from the domain of the quotient function.
Precalculus math problems on this topic will look something like the following:
Sample Precalculus Function Problem #1
For the indicated function f and g, find the functions f+g, f-g, fg, and f/g, and find their domains.
f(x) = 4x; g(x) = x+1
Answer:
(f+g)(x) = f(x) + g(x) = 4x + x + 1 = 5x + 1
(f-g)(x) = f(x) – g(x) = 4x – (x + 1) = 4x – x – 1 = 3x – 1
(fg)(x) = f(x)g(x) = 4x(x+1) = 4×2+4x
The domain for all of the above is all Real numbers.
(f/g)(x) = f(x)/g(x) = 4x/(x+1)
The domain for the Quotient function in this case is all Real numbers except -1. You get this value by looking at the denominator. Since you can not divide by 0, x=-1 is not a legal domain value for this function.
Sample Precalculus Function Problem #2
For the indicated function f and g, find the functions f+g, f-g, fg, and f/g, and find their domains.
f(x) = 2×2; g(x) = x2 + 1
Answer:
(f+g)(x) = f(x) + g(x) = 2×2 + x2 + 1 = 3×2 + 1
(f-g)(x) = f(x) – g(x) = 2×2 – (x2 + 1) = x2 – 1
(fg)(x) = f(x)g(x) = 2×2( x2 + 1) = 2×4 + 2×2
(f/g)(x) = f(x)/g(x) = 2×2/(x2 + 1)
The domain for all functions in this case is all Real numbers.
Blessings!
Source
Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen. Precalculus. Functions and Graphs. Fifth Edition