(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose [itex]f[/itex] is one element of [itex]\mathbb{A}[/itex], and it has the property that [itex]f \circ g = g \circ f[/itex] for every [itex]g \in \mathbb{A}[/itex]. Prove that [itex]f = e[/itex] (the identity function).

2. Relevant equations

[itex]\mathbb{A} = \{ g_{ab} : (a, b) \in \mathbb{R}^2, \, a \neq 0 \}[/itex]

[itex]g_{ab}(x) = ax + b[/itex]

3. The attempt at a solution

Choose [itex]f, g \in \mathbb{A}[/itex]. By definition, [itex]f[/itex] and [itex]g[/itex] have the form [itex]g_{ab}[/itex] and [itex]g_{cd}[/itex] for some [itex](a, b) \in \mathbb{R}^2[/itex] and [itex](c, d) \in \mathbb{R}^2[/itex] such that [itex]a \neq 0[/itex] and [itex]c \neq 0[/itex].

Since [itex]f \circ g = g \circ f[/itex], we find that

[itex]f(g(x)) = g(f(x))[/itex]

[itex]f(cx + d) = g(ax + b)[/itex]

[itex]a(cx + d) + b = c(ax + b) + d[/itex]

[itex]acx + ad + b = cax + cb + d[/itex].

This reduces to [itex]ad + b = cb + d[/itex].

This is where I get stuck. I realize that we are working towards finding [itex]a = 1[/itex] and [itex]b = 0[/itex], so that [itex]g_{ab} = g_{10} = x[/itex] (the identity function). Thank you for your help!

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Prove a function is the identity

**Physics Forums | Science Articles, Homework Help, Discussion**