Basic function operations
There are four basic function operations:
- Addition
- Subtraction
- Multiplication
- Division
Addition
- Because of the commutative property of addition, it does not matter in what order we add the functions.
Let's add f(x)=x-1 and f(x)=x²-2x+1
We can begin by lining up like terms:
x-1
+ x²-2x+1
x²-x
f(x)= x²-x is our final answer. We can put our answer in terms of f(x) because we know it is a function. We can look at a graph and see that the graph would pass the vertical line test, or look at a table or map of values and see that there is only one output for each input. (f(x)= x²-x is graphed in purple below.)
Pretty easy. Let's try f(x)=x³-1 and f(x)=x-1
x³+0x²+0x-1
+ x-1
x³ +x-2
By adding these two functions, we come up with f(x)=x³+x-2. This is also a function; f(x)=x³+x-2 is graphed in green below.
Subtraction
- The order of the functions for subtraction does matter.
f(x)=x²-2x+1 minus f(x)=x-1
x²-2x+1
- x -1
x²-3x+2
f(x)= x²-3x+2 is graphed in blue on the graph to the right.
f(x)=x-1 minus f(x)=x³-1
0x³+x-1
- x³+0x-1
-x³+x
f(x)= -x³+x is graphed in orange on the graph to the right.
x²-2x+1
- x -1
x²-3x+2
f(x)= x²-3x+2 is graphed in blue on the graph to the right.
f(x)=x-1 minus f(x)=x³-1
0x³+x-1
- x³+0x-1
-x³+x
f(x)= -x³+x is graphed in orange on the graph to the right.
Multiplication
- Because of the commutative property of multiplication, it does not matter in what order we multiply the functions.
Let's multiply f(x)=x-1 and f(x)=x-1:
(x-1)(x-1)
x²-x-x+1
x²-2x+1
f(x)=x²-2x+1 is our solution. It is a function; it is graphed in purple on the graph to the left.
Let's multiply f(x)=x-1 and f(x)=x²-2x+1
(x-1)( x²-2x+1)
x³-2x²+x-x²+2x-1
x³-3x²+3x-1
f(x)=x³-3x²+3x-1 is the solution. This function is graphed in red on the graph to the left.
(x-1)(x-1)
x²-x-x+1
x²-2x+1
f(x)=x²-2x+1 is our solution. It is a function; it is graphed in purple on the graph to the left.
Let's multiply f(x)=x-1 and f(x)=x²-2x+1
(x-1)( x²-2x+1)
x³-2x²+x-x²+2x-1
x³-3x²+3x-1
f(x)=x³-3x²+3x-1 is the solution. This function is graphed in red on the graph to the left.
Division
- The order of the functions in division does matter, and will alter the answer.
- Order of the functions does matter for division.
Let's divide f(x)=x²-2x+1 by f(x)=x-1. We will solve this problem both ways; we should obtain the same answer either way.
x - 1
x-1/ x²-2x+1 - x² - x -x + 1 - -x + 1 0 |
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Try applying the following examples, then click the "Answer" button to check your work.
f(x)=x³-1 plus f(x)=x³-1
f(x)=x²-2x+1 minus f(x)=x³-1
f(x)=x-1 times f(x)=x²-1
f(x)=x³+8 divided by f(x)=x²-3x-10
f(x)=x³-1 plus f(x)=x³-1
f(x)=x²-2x+1 minus f(x)=x³-1
f(x)=x-1 times f(x)=x²-1
f(x)=x³+8 divided by f(x)=x²-3x-10