Evaluating & Composing
Composition means feeding the output of one function into another, written f(g(x)) and read "f of g of x". The order matters and it runs inside-out: to compute f(g(2)), first evaluate g(2), then put that number into f. Reversing the order — computing g(f(2)) instead — is the single most common composition mistake, and the GRE builds trap answers around it. Work numerically whenever you can (find g(2) as an actual number, then apply f) rather than building the full algebraic formula for f(g(x)), which is slower and error-prone under time pressure. When the problem does want the general composite expression, substitute the entire g(x) rule into f. Composition also appears with tables or graphs: read g(2) off the table, then look up f of that value. In every case, resist the urge to combine the rules by adding or multiplying — composition is substitution, layered.
✅ Solved examples
✏️ Practice — try these, take hints as needed
📝 Topic test — 8 questions
Auto-graded with full solutions; saved to your dashboard. Use the calculator and formula sheet (top-right) any time.
Formula Reference Sheet
Notation & evaluation
| Function value | f(a) = replace every x in the rule with a |
|---|---|
| Composition | f(g(x)) = evaluate g first, then feed into f |
| Custom operator | a ⊕ b = (whatever rule the problem defines) |
Domain & range
| Domain | all inputs x that are allowed |
|---|---|
| Range | all outputs f(x) that can result |
| Denominator rule | exclude any x making a denominator 0 |
| Even-root rule | the quantity under a √ must be ≥ 0 |