Consider the function $g(x) = 2 \sin^{-1}(3x)$ - HSC - SSCE Mathematics Extension 1 - Question 5 - 2024 - Paper 1

To analyze the transformations from $f(x) = \sin^{-1}(x)$ to $g(x) = 2 \sin^{-1}(3x)$, we can break it down into vertical and horizontal transformations:

1. Vertical Transformation:

   • The presence of $2$ in $g(x)$ indicates a vertical dilation. This means that each output of $f(x)$ is scaled by a factor of $2$.

2. Horizontal Transformation:

   • The argument $3x$ inside the function suggests a horizontal dilation. The factor affecting the input $x$ is $3$, which means we are compressing the function horizontally by a factor of rac{1}{3} since a larger coefficient in the input scales the graph horizontally.

In conclusion, the transformations are:

• Vertical dilation by a factor of $2$.
• Horizontal dilation by a factor of $\frac{1}{3}$.