Composition of two functions is always commutative.Group startsTrue or FalseTrue, unselectedFalse, unselected

Break Down the Problem

1. Understand what function composition is.
2. Assess whether composition of functions satisfies the commutative property.

Relevant Concepts

1. Function Composition: If $f$ and $g$ are two functions, their composition is given by $(f \circ g)(x) = f(g(x))$.
2. Commutative Property: A binary operation $\ast$ is commutative if $a \ast b = b \ast a$ for any $a$ and $b$.

Analysis and Detail

1. To determine if composition is commutative, we need to check whether $f(g(x)) = g(f(x))$ holds true for all functions $f$ and $g$.

2. Taking two specific functions as examples:

   • Let $f(x) = x + 1$
   • Let $g(x) = 2x$

   Evaluating $f \circ g$: $(f \circ g)(x) = f(g(x)) = f(2x) = 2x + 1$

   Now evaluating $g \circ f$: $(g \circ f)(x) = g(f(x)) = g(x + 1) = 2(x + 1) = 2x + 2$

   Since $2x + 1 \neq 2x + 2$, it shows that if $f$ and $g$ are chosen this way, the compositions are not equal.

Verify and Summarize

1. Since we have found a counterexample to the claim of commutativity, we can conclude that function composition is not always commutative.

Final Answer

False: The composition of two functions is not always commutative.