Q 116. - Precalculus Enhanced with Graphing Utilities

Question

If $f\left(x\right)=\sin x ,g\left(x\right)= \cos x,$and $h\left(x\right)=2x ,P\left(x\right)=\frac{x}{2}$then find the values of points given below.

(a) Find $g\left(\frac{\pi }{6}\right)$.What point is on the graph of g.

(b) Assuming g  one to one use the result of part (a) to find a point on the graph of $g^{-1}$.

(c) What point is on the graph of $y=2g\left(x-\frac{\pi }{6}\right)$ if $x=\frac{\pi }{6}$

Step-by-Step Solution

Answer

(a) Value of $g\left(\frac{\pi }{6}\right)$ is $\frac{\sqrt{3}}{2}$.

(b) $g^{-1}=\frac{2}{\sqrt{3}}$

(c) $y=0$

Step 1. Given information .

Consider the given value $g\left(x\right)=\cos x$ where $x=\frac{\pi }{6}$ .

Step 2. Finding g = π 6 .

Substitute $g\left(\frac{\pi }{6}\right)=\cos \frac{\pi }{6}$in function $g\left(x\right)=\cos x$. In trigonometric standard angle the value of $\cos \frac{\pi }{6}=30°=\frac{\sqrt{3}}{2}$so,

$g\left(\frac{\pi }{6}\right)=\cos \frac{\pi }{6}=\frac{\sqrt{3}}{2}$

Step 3. Plot the graph.

The graph for $g\left(\frac{\pi }{6}\right)$ is shown below.

Step 4. (b) Finding g - 1 .

$$\begin{gathered} g^{-1}=\frac{1}{g} \\ =\frac{1}{{\displaystyle \frac{\sqrt{3}}{2}}} \\ =\frac{2}{\sqrt{3}} \end{gathered}$$

Step 5.(c) Finding a point on the graph .

Substitute the value of $x$ in function $y=2g\left(x-\frac{\pi }{6}\right)$

$$\begin{gathered} y=2\times g\left(\frac{\pi }{6}-\frac{\pi }{6}\right) \\ y=0 \end{gathered}$$

The graph for $y=0$ shown below.