In the coordinate plane, let $f$ be the rotation transformation that rotates by $\frac { \pi } { 3 }$ about the origin, and let $g$ be the reflection transformation about the line $y = x$. When the line $x + 2 y + 5 = 0$ is mapped to the line $a x + b y + 5 = 0$ by the composite transformation $g ^ { - 1 } \circ f \circ g$, what is the value of $a + 2 b$? (Given that $a , b$ are constants.) [3 points] (1) $\frac { 1 } { 2 }$ (2) 1 (3) $\frac { 3 } { 2 }$ (4) 2 (5) $\frac { 5 } { 2 }$
In the coordinate plane, let $f$ be the rotation transformation that rotates by $\frac { \pi } { 3 }$ about the origin, and let $g$ be the reflection transformation about the line $y = x$. When the line $x + 2 y + 5 = 0$ is mapped to the line $a x + b y + 5 = 0$ by the composite transformation $g ^ { - 1 } \circ f \circ g$, what is the value of $a + 2 b$? (Given that $a , b$ are constants.) [3 points]\\
(1) $\frac { 1 } { 2 }$\\
(2) 1\\
(3) $\frac { 3 } { 2 }$\\
(4) 2\\
(5) $\frac { 5 } { 2 }$