Q. 10 - Precalculus Enhanced with Graphing Utilities

Question

g$\left(x\right)$ = cos x

(a) What is the y-intercept of the graph of g?

(b) For what numbers x, $-\mathrm{\pi }\le \mathrm{x}\le \mathrm{\pi }$, is the graph of g

decreasing?

(c) What is the absolute minimum of g?

(d) For what numbers x,$0\le x\le 2\mathrm{\pi }$, does g$\left(x\right)=0$?

(e) For what numbers x, $-2\mathrm{\pi }\le \mathrm{x}\le 2\mathrm{\pi }$, does g(x) = 1?

Where does g(x) = -1?

(f) For what numbers x, $-2\mathrm{\pi }\le \mathrm{x}\le 2\mathrm{\pi }$, does g(x)=$\frac{\sqrt{3}}{2}$?

(g) What are the x-intercepts of g ?

Step-by-Step Solution

Answer

(a) The y-intercept is $1.$

(b) 

for decreasing function $x\in \left[0,\mathrm{\pi }\right]$

(c) The absolute minimum of g is $-1$.

(d) $g\left(x\right) = 0 for x\in \{-\frac{3\mathrm{\pi }}{2}-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2},\frac{3\mathrm{\pi }}{2}\}$

(e) $$\begin{gathered} g\left(x\right)=1 for \\ x=-2\mathrm{\pi },\mathrm{x}=0 \mathrm{and} x=2\mathrm{\pi } \\ g\left(x\right)=-1 for \\ x=-\mathrm{\pi } and x=\mathrm{\pi } \end{gathered}$$

(f) $$\begin{gathered} f\left(x\right)=\frac{\sqrt{3}}{2} for \\ x=-\frac{11\mathrm{\pi }}{6},-\frac{\mathrm{\pi }}{6},\frac{\mathrm{\pi }}{6},\frac{11\mathrm{\pi }}{6} \end{gathered}$$

(g) The x intercept is $(2k+1)\frac{\mathrm{\pi }}{2} where k is an integer.$

Part (a) Step 1. Given Information

$g\left(x\right) = \cos x$

Part (a) Step 2. Calculation

Since the graph of $g\left(x\right)=\cos x$ passes through (0,1) therefore the y-intercept of the graph is 1.

Part (b) Step 1. Calculation

The function $g\left(x\right)=\cos x$ is decreasing function $from x=\left[0,\mathrm{\pi }\right]$ 

Part (c) Step 1. Calculation

The amplitude of the $g\left(x\right)=\cos x is \pm 1$ therefore the minimum value is -1.

Part (d) Step 1. Calculation

$$\begin{gathered} g\left(x\right) = 0 \\ for x\in \{-\frac{3\mathrm{\pi }}{2}-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2},\frac{3\mathrm{\pi }}{2}\} \end{gathered}$$

Part (e) Step 1. Calculation

$$\begin{gathered} g\left(x\right)=1 for \\ x=-2\mathrm{\pi },\mathrm{x}=0 \mathrm{and} x=2\mathrm{\pi } \\ g\left(x\right)=-1 for \\ x=-\mathrm{\pi } and x=\mathrm{\pi } \end{gathered}$$

Part (f) Step 1. Calculation

$$\begin{gathered} f\left(x\right)=\frac{\sqrt{3}}{2} for \\ x=-\frac{11\mathrm{\pi }}{6},-\frac{\mathrm{\pi }}{6},\frac{\mathrm{\pi }}{6},\frac{11\mathrm{\pi }}{6} \end{gathered}$$

Part (g) Step 1. Calculation

The x intercept is $(2k+1)\frac{\mathrm{\pi }}{2} where k\in I.$