Question

In Problems 67–78,

$f\left(x\right)=\sin x,-\frac{\mathrm{\pi }}{2}\le x\le \frac{\mathrm{\pi }}{2},g\left(x\right)=\cos x,0\le x\le \mathrm{\pi }\mathrm{and}\mathrm{h}\left(\mathrm{x}\right)=\mathrm{tanx},-\frac{\mathrm{\pi }}{2}<x<\frac{\mathrm{\pi }}{2}$

Find the exact value of each composite function.

$h\left(g^{-1}\left(-\frac{1}{4}\right)\right)$

Short answer

The required exact value of given expression is$-\sqrt{15}$.

Step-by-step solution

Step 1. Given information. 

The composite function$h\left(g^{-1}\left(-\frac{1}{4}\right)\right)$.

Step 2. Here, let g-1-14=x

So,

$$\begin{gathered} g\left(x\right)=-\frac{1}{4} \\ \cos x=-\frac{1}{4} \\ Since,\cos x<0,so\frac{\mathrm{\pi }}{2}\le x\le \mathrm{\pi } \\ \mathrm{Now}, \\ \mathrm{sinx}=\sqrt{1-{\cos }^2\mathrm{x}} \\ =\sqrt{1-{\left(-\frac{1}{4}\right)}^2} \\ =\sqrt{1-\frac{1}{16}} \\ =\sqrt{\frac{15}{16}} \\ =\frac{\sqrt{15}}{4} \end{gathered}$$

Step 3. Now, find hg-1-14.

$$\begin{gathered} h\left(g^{-1}\left(-\frac{1}{4}\right)\right)=h\left(x\right) \\ =\tan x \\ =\frac{\sin x}{\cos x} \\ =\frac{\sqrt{15}/4}{-1/4} \\ =-\sqrt{15} \end{gathered}$$