Answers: addition, subtraction
The commutative property holds for the addition of functions, but not for the subtraction of functions.
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Step-by-step explanation:
Let's say we wanted to add two numbers 3 and 5. Doing so gets us 3+5 = 8. We could also easily say 5+3 = 8. The order of addition doesn't matter. So the commutative property holds for addition.
Unfortunately, the same cannot be said for subtraction. Note that 5-3 = 2 while 3-5 = -2. We can think of it in terms of money: Let's say we have $5 in our pocket and buy an item worth $3. That would leave 5-3 = 2 dollars left over. If we flip things around and have $3 in our pocket, and buy something worth $5, then 3-5 = -2 indicates we're $2 in debt. This is one example to help see why negative numbers are useful.
In short, subtraction is not commutative. In general, x-y is not the same as y-x.
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All of what was discussed in the previous section only talks about single numbers, instead of functions. The good news is that functions are simply extensions of numbers in a way. Rather than just talking about one number, we're talking about a collection of them. The same idea applies:
The order of adding functions doesn't matter, so addition is commutative for functions. However, the order of subtraction does matter for functions. So function subtraction is not commutative.
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An example:
f(x) = x+5
g(x) = 2x-3
f - g = (x+5) - (2x-3) = x+5-2x+3 = -x+8
g - f = (2x-3) - (x+5) = 2x-3 - x-5 = x - 8
The results -x+8 and x-8 are two different expressions. You can form a table of xy values, or use a graph, to see that they are different expressions