Common Mistakes

Frequent Errors

1. Computing $f(x)\cdot g(x)$ instead of $f(g(x))$. 2. Reversing order: using $(g\circ f)(x)$ when $(f\circ g)(x)$ is required. 3. Ignoring domain: $g(x)$ must land in $\text{Dom}(f)$. 4. Assuming $f\circ g=g\circ f$ (2.7.A.3). 5. Decomposition that does not recompose to the original. 6. Forgetting parentheses when substituting: $(x+3)^2$ vs $x^2+3$. 7. Table/graph chain reading outer before inner. 8. Treating $f\circ I$ or $I\circ f$ incorrectly (identity gives $f$).

Why These Happen

Students confuse composition with multiplication and struggle with nested input-output flow.

How to Fix Them

• Write "Step 1: $u=g(x)$. Step 2: $f(u)$."
• Domain template on every composite.
• One counterexample where $f(g(1))\neq g(f(1))$.
• Recomposition check after decomposing.

Avoidance Scripts

• "Inside first; outside second."
• "$f\circ g$ means $f$ receives $g$'s output."
• "Test your decomposition by composing back."

Quick Self-Check

• Did I use the correct order?
• Did I check domain?
• Does my decomposition recompose correctly?