Which diagram represents the domain of the function $f(x) = ext{sin}^{-1}igg(rac{3}{x}igg)$? A - HSC - SSCE Mathematics Extension 1 - Question 7 - 2017 - Paper 1

1. For the left side:

   $-1 \leq \frac{3}{x}$ Rearranging gives:

   $x \leq -3$ (since $x$ must be negative for it to hold true)

2. For the right side:

   $\frac{3}{x} \leq 1$ Rearranging gives:

   $x \geq 3$ (since $x$ must be positive for this to hold true)

Thus, the domain of the function includes values of $x$ in the intervals:

• $(-\infty, -3]$
• $[3, \infty)$

The correct representation of these intervals is reflected by the option that shows $(-\infty, -3]$ and $[3, \infty)$ which corresponds to answer A.