Function Notation and Composition: formulas you must memorize

Why memorize when the SAT lists formulas?

The SAT Math section opens with a reference sheet that lists a handful of geometry formulas. Most students glance at it once and never look again. The reason is simple: looking up a formula mid-section costs five to ten seconds, and across the dozens of questions that need a formula, those seconds compound into a full question's worth of lost time. Memorizing every formula on this page eliminates that drag.

This page lists every formula you should have at automatic recall for Function Notation and Composition questions, with a short note on when each one applies and which question patterns reliably use it.

The formulas

1. f(g(x)) means apply g first, then f

   This formula is the workhorse for at least one of the four function notation and composition sub-skills ScoreReady drills. You will see it appear in the worked solutions on the topic practice page across multiple difficulty levels.

2. Inverse: swap x and y, then solve for y

   This formula is the workhorse for at least one of the four function notation and composition sub-skills ScoreReady drills. You will see it appear in the worked solutions on the topic practice page across multiple difficulty levels.

3. f(f^(−1)(x)) = x

   This formula is the workhorse for at least one of the four function notation and composition sub-skills ScoreReady drills. You will see it appear in the worked solutions on the topic practice page across multiple difficulty levels.

4. f(c) is the y-value when x = c

   This formula is the workhorse for at least one of the four function notation and composition sub-skills ScoreReady drills. You will see it appear in the worked solutions on the topic practice page across multiple difficulty levels.

How to memorize

The fastest reliable way to memorize a small set of formulas is spaced repetition. On day one, write the formula and its trigger phrase three times each. On day two, cover the formula and recall it from the trigger phrase alone. On day four, do the same. On day eight, drill again. After four cycles spaced this way, the formulas are durable for at least six weeks — long enough to cover most SAT preparation timelines.

An equivalent method that some students prefer is to embed the formula inside a worked example. Write a single SAT-style problem that uses the formula, work it on paper end to end, then re-derive the formula from the example whenever you need it. This anchors the abstract symbols to a concrete pattern.

Common application errors

• Computing g(f(x)) when the question asks for f(g(x))
• Reflecting across the wrong axis when sketching an inverse
• Missing one occurrence of x when substituting an algebraic input
• Reading the x-coordinate when the question asks for f(c)

Each of these errors corresponds to a misapplication of one of the formulas above. Read the pitfalls walkthrough for a worked example of each, then drill the topic question bank with conscious attention to which pitfall each question is testing.