Question

If the co-domain of the function \[ f(x)= \begin{cases} 3\sin x-4\cos x, & x\in \left(\tan^{-1}\frac{4}{3}-\frac{\pi}{2},\,\tan^{-1}\frac{4}{3}+\frac{\pi}{2}\right)\\[2mm] \log(\sin x), & x\in \left(\frac{5\pi}{6},\pi\right) \end{cases} \] is \((-\infty,5)\), then \(f\) is

• A. an injection but not a surjection
• B. a surjection but not an injection
• C. a bijection
• D. neither an injection nor a surjection

Hint:

For piecewise functions, first find the range of each branch separately and then take their union. Compare the resulting range with the given co-domain to test surjectivity.

Answer 1

The Correct Option is B