Question

Let $f:R\to \left(0,\frac{\pi }{6}\right]$ be defined as $f(x)={\sin }^{-1}\left(\frac{4}{4x^2-12x+17}\right)$ then $f(x)$ is

• A. injective as well as surjective.
• B. surjective but not injective.
• C. injective but not surjective.
• D. neither injective nor surjective.

Answer 1

Answer: (B)

We have $f(x)={\sin }^{-1}\left(\frac{4}{4x^2-12x+17}\right)$
$={\sin }^{-1}\left(\frac{1}{x^2-3x+\frac{17}{4}}\right)={\sin }^{-1}\left(\frac{1}{{\left(x-\frac{3}{2}\right)}^2+\frac{17}{4}-\frac{9}{4}}\right)={\sin }^{-1}\left(\frac{1}{{\left(x-\frac{3}{2}\right)}^2+2}\right)$
Hence $f(x)\in \left(0,\frac{\pi }{6}\right]$, so co-domain $=$ Range $($ As $x^2-3x+\frac{17}{4}={\left(x-\frac{3}{2}\right)}^2+2\in [2,\infty ))$
Also $y=4x^2-12x+17$ is many one function.
Hence $f(x)$ is surjective but not injective.