ACT® · Math: Functions — Properties of Functions
Made-Up Operations Such as a diamond b = 3a - b (AF 304)
A made-up operation like a diamond b = 3a - b is just a function with two inputs: whatever sits before the symbol goes in for a, whatever sits after goes in for b, in that exact order.
The test can invent any symbol it wants, a diamond, a star, a box, because it always hands you the definition in the same breath. You're never expected to know what the symbol means ahead of time. You're being tested on whether you can follow a rule mechanically: read the definition once, then substitute. That's why it can be solved fast. It's not a new math concept, it's a plug-in exercise wearing a costume.
Read a diamond b = 3a - b as a function of two inputs: the first slot (a) gets multiplied by 3, the second slot (b) gets subtracted off. Whatever is written immediately to the left of the diamond becomes a, and whatever is to the right becomes b. If the right-hand side is a whole expression, like (q - x), treat it as one block and subtract the entire thing, not just part of it. Keep parentheses around it until you've simplified.
Because the two slots play different roles, order matters. a diamond b is 3a - b, but b diamond a is 3b - a, a completely different expression unless a and b happen to make them equal. There's no rule saying invented operations have to be commutative like addition or multiplication are. This connects to function notation generally: a diamond b is really f(a,b), and the same care you'd use plugging numbers into f(x) applies here, just with two slots instead of one.
Key terms
- Made-up operation
- A symbol invented just for one problem, defined on the spot and never used anywhere else.
- Two-input function
- A rule that takes two separate values and combines them into one output, like a diamond b = 3a - b.
- Commutative
- A property where swapping the order of two inputs gives the same result, true for addition but not guaranteed for invented operations.
- Slot
- A position in the operation's definition, first or second, that determines what role a given input plays.
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End of topic.