Q 115. - 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 value of each of the following:

(a) Find $f\left(\frac{\pi }{4}\right)$ .What point is on the graph of f ?

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

(c) What  point is on the graph of $y=f\left(x+\frac{\pi }{4}\right)-3$ if $x=\frac{\pi }{4}$.

Step-by-Step Solution

Answer

(a) $f\left(\frac{\pi }{4}\right)$ is $\frac{1}{\sqrt{2}}$.

(b) $f^{-1}=\sqrt{2}$

The graph for $f\left(\frac{\pi }{4}\right)$ is,

Step 1. Given information.

Consider the given value $f\left(x\right)=\sin x$ where $x=\frac{\pi }{4}$

Step 2. Finding f π 4 .

Substitute $f\left(\frac{\pi }{4}\right)=\sin \frac{\pi }{4}$ in the  trigonometric standard angle the value of $\sin \frac{\pi }{4}=45°=\frac{1}{\sqrt{2}}$

Step 3. Finding f - 1 .

Substitute the value .

$$\begin{gathered} f^{-1}=\frac{1}{f}=\frac{1}{\sin {\displaystyle \frac{\pi }{4}}} \\ =\frac{1}{{\displaystyle \frac{1}{\sqrt{2}}}} \\ =\sqrt{2} \end{gathered}$$

Step 4. Finding a point on the graph.

To find a point on the graph substitute the value $x=\frac{\pi }{4}$in the function,

$$\begin{gathered} y=f\left(x+\frac{\pi }{4}\right)-3 \\ y=\sin \frac{2\pi }{4}-3 \\ y =-2 \end{gathered}$$

Step 5. plot the graph.

The graph for the equation $y=-2$ .