ACT and SAT Math Tips: Symbols and Functions

When I teach ACT and SAT classes, I find that the most intimidating math topic is always the Symbolic Function. The complaints are inevitable: “But I don’t know what an upside down triangle even means!” or “I’ve never seen this in school!” or “Does a star inside of a circle mean multiply? I just don’t understand!”

We are programmed to believe that symbols in math demand a specific action from us (multipy, divide, add, subtract, square, etc.), so it’s difficult for us to accept the truth about symbolic functions: the symbols themselves do not signal any definitive operation. They are simply an announcement, yelling, “Hey, I’m a function!” If a symbolic function uses a diamond, it doesn’t matter if I replace the diamond with a circle, star, square, or four-leaf clover; the question will be solved exactly the same way, no matter what symbol is chosen to announce the presence of a function.

Let’s look at an example:

It may help to think of all function questions as having a ‘puzzle’ and a ‘key.’ The puzzle involves the original equation:

The key is the portion of the question that places new expressions or terms into the function:

To solve these questions, align the key directly below the puzzle:

The key reveals that x = 4 and y = 8. In the puzzle, substitute a 4 for every x and an 8 for every y:

Then solve the equation:

The correct answer is (C).

Alignment is extremely important, especially when the key involves expressions:

For every g, substitute the expression x + 3. For every h, substitute x – 4. Use parentheses for the expressions so that you remember to distribute any negatives or exponents:

Using proper alignment can save you from careless mistakes on the ACT and SAT.

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Photo: “Signs_Symbols,” courtesy of Liga Eglite