Sets, Relations & Functions
Domain of sin⁻¹ of Log Function — Bonus
nta_pyq_2023_apr
Grade 11
Question:
Let $D$ be the domain of $f(x)=\sin^{-1}\!\left(\log_{3x}\dfrac{6+2\log_{3x}}{-5x}\right)$. If the range of $g:D\to\mathbb{R}$ defined by $g(x)=x-[x]$ is $(\alpha,\beta)$, then $\alpha^2+\dfrac{5}{\beta}$ is equal to
$135$ (approx)
$2$
$3$
$\dfrac{3}{2}$
Step-by-Step Solution
Key Concept: Find domain by requiring $3x>0$, $3x\neq1$, argument $>0$, and the argument in $[-1,1]$. The domain is approximately $(0,\frac{1}{27})$.
Bonus question. $\alpha\approx0,\ \beta\approx\frac{1}{27}$. $\alpha^2+\frac{5}{\beta}\approx135$.
Correct Answer: 1