Question

For \( a, \beta \in \mathbb{R} \), define the map \( \varphi_{a,\beta}: \mathbb{R} \to \mathbb{R} \) by \[ \varphi_{a,\beta}(x) = ax + \beta. \] Let \[ G = \{ \varphi_{a,\beta} \mid (a, \beta) \in \mathbb{R}^2 \}. \] For \( f, g \in G \), define \( g \circ f \in G \) by \[ (g \circ f)(x) = g(f(x)). \] Then which of the following statements is/are TRUE?

• A. The binary operation \( \circ \) is associative.
• B. The binary operation \( \circ \) is commutative.
• C. For every \( (a, \beta) \in \mathbb{R}^2, a \neq 0 \), there exists \( (a', \beta') \in \mathbb{R}^2 \) such that \( \varphi_{a,\beta} \circ \varphi_{a',\beta'} = \varphi_{1,0} \).
• D. \( (G, \circ) \) is a group.

Hint:

The composition of linear maps is associative, but not necessarily commutative. Additionally, for non-zero \( a \), an inverse map always exists to satisfy \( \varphi_{a,\beta} \circ \varphi_{a',\beta'} = \varphi_{1,0} \).

Answer 1

The Correct Option is A, C

Step 1: Analyzing associativity.
The operation \( \circ \) is associative if for all \( f, g, h \in G \), we have: \[ (f \circ (g \circ h)) = ((f \circ g) \circ h). \] Since \( g \circ f \) simply involves function composition, associativity holds for composition of linear maps, as function composition is always associative.
Step 2: Analyzing commutativity.
The operation \( \circ \) is not commutative because, in general, \( g(f(x)) \neq f(g(x)) \). Therefore, option (B) is not true.
Step 3: Analyzing the existence condition.
We check whether there exists \( (a', \beta') \in \mathbb{R}^2 \) such that \( \varphi_{a,\beta} \circ \varphi_{a',\beta'} = \varphi_{1,0} \). Solving this equation: \[ \varphi_{a,\beta}(x) = ax + \beta, \quad \varphi_{a',\beta'}(x) = a'x + \beta', \] we need \( \varphi_{a,\beta} \circ \varphi_{a',\beta'}(x) = \varphi_{1,0}(x) = x \), which gives the solution \( a = 1/a' \) and \( \beta = -\beta' \). Therefore, such a pair \( (a', \beta') \) exists for every \( (a, \beta) \in \mathbb{R}^2 \), where \( a \neq 0 \).
Step 4: Conclusion.
Thus, the correct answers are (A) and (C).