Overview

A defined operation problem will usually give you a made up mathematical operation defined through a custom symbol, such as $a \star b$ or $a \diamond b$, and tell you what it equals in terms of $a$ and $b$. Then, the problem will ask you to evaluate a specific expression or determine the value of a number $a$ or $b$ such that an expression involving the defined operation holds true. These problems test careful substitution and order of operations so it is important that you ensure accurate calculations.

Key Ideas

• Treat the defined symbol as a function of its inputs.
• Substitute the given values exactly where the definition tells you to.
• If the operation is nested (e.g., $2 \star (3 \star 4)$), evaluate the inner operation first.
• Watch for non-commutativity: $a \star b$ may differ from $b \star a$.

Core Skills

Direct Substitution

Direct substituion is simply solving a defined operation problem by substituting the given values into the operation function.

Demo Problem: Given that $a \diamond b = 2a + b^2$, find $3 \diamond 4$.

We solve this problem by directly substituting in $a=3$ and $b=4$:

It is important to ensure that we don't get $a$ and $b$ messed up and use $a=4$ and $b=3$.

Nested Operations

When nested operations are involved in a given expression, ensure that you first compute the inner result, and then feed it into the outer operation.

Demo Problem: Given that $a \oplus b = (2a)^2 + 3b$, find $3 \oplus (4 \oplus 5)$.

We solve this problem by first evaluating $(4 \oplus 5)$, and then using that value to evaluate $3 \oplus (4 \oplus 5).$

Finding Unknowns

We often asked to determine the value of unknown variables used in defined operation expressions.

Demo Problem: If $x \star 5 = 20$ and $a \star b = ab - a$, determine the value of $x$.

It suffices to substitute $a=x$ and $b=5$ and solve the resulting algebraic equation.

Practice Problems