Assuming fg(x) equals gf(x) H
"The Commutative Confusion"
The Mistake in Action
Given $f(x) = x^2$ and $g(x) = x + 1$.
Student states: "$fg(x) = gf(x)$ because multiplication is commutative."
Why It Happens
Students apply the commutative property of multiplication to function composition, which doesn't work.
The Fix
Function composition is NOT commutative: $fg(x) \neq gf(x)$ in general.
$fg(x) = f(g(x)) = f(x + 1) = (x + 1)^2 = x^2 + 2x + 1$
$gf(x) = g(f(x)) = g(x^2) = x^2 + 1$
These are different! The order matters because different operations are being done.
Think of it like getting dressed: Socks then shoes ≠ Shoes then socks!
Spot the Mistake
Can you identify where this student went wrong?
$fg(x) = gf(x)$ because multiplication is commutative
Click on the line that contains the error.
Related Topics
Learn more about the underlying maths: