Functions • Topic 4 of 4

Custom (Symbol) Functions

A custom-operator question invents a symbol you have never seen — ◊, ⊕, ∇, a boxed number — and defines what it does, then asks you to compute. This is a signature GRE format and it is engineered to look scary. It is not: the definition IS the rule, and your only job is careful substitution. If the problem says "for all numbers x and y, x ◊ y = x² − 2y", then 5 ◊ 3 = 5² − 2(3) = 25 − 6 = 19. The traps are ordinary: watch the order of the two inputs (x ◊ y need not equal y ◊ x), respect the exponents and signs, and when a symbol is nested — like (2 ◊ 3) ◊ 4 — evaluate the inner bracket to a number first, then apply the rule again. Some versions define the operator on a single input (⌊x⌋ or "x* = x + 1") or ask you to solve for the unknown input that makes the expression equal a target. Read the definition twice, substitute in parentheses, and these become some of the most reliable points on the section.

✅ Solved examples

1. Define x ◊ y = 2x + y. Find 4 ◊ 5.
Substitute x = 4, y = 5: 2(4) + 5 = 8 + 5 = 13.
2. Define a ⊕ b = a² − b. Show 3 ⊕ 5 and 5 ⊕ 3 are different.
3 ⊕ 5 = 3² − 5 = 9 − 5 = 4, while 5 ⊕ 3 = 5² − 3 = 25 − 3 = 22. Order matters — the operator is not symmetric.
3. Define x ∇ y = xy − x. Find (2 ∇ 3) ∇ 4.
Inner: 2 ∇ 3 = (2)(3) − 2 = 4. Then 4 ∇ 4 = (4)(4) − 4 = 12.
4. Define x ★ y = x + 2y. If 6 ★ n = 20, find n.
Set up the rule: 6 + 2n = 20, so 2n = 14 and n = 7.

✏️ Practice — try these, take hints as needed

1. Define x ◊ y = 3x − y. Find 7 ◊ 4.
Substitute x = 7, y = 4.
3(7) − 4.
21 − 4.
17
2. Define a ⊕ b = a + b². Find 5 ⊕ 2.
b is squared, not a.
5 + 2².
5 + 4.
9
3. Define x ∇ y = 2xy. Find (1 ∇ 2) ∇ 3.
Evaluate the inner bracket first.
1 ∇ 2 = 2(1)(2) = 4.
Then 4 ∇ 3 = 2(4)(3).
24
4. Define x ⊙ y = x² + y². Find 3 ⊙ 4.
Square each input.
9 + 16.
Add.
25
5. Define x ▲ y = 4x − 3y. If 5 ▲ k = 11, find k.
Write 4(5) − 3k = 11.
20 − 3k = 11.
3k = 9.
k = 3

📝 Topic test — 8 questions

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