Let's analyze and solve the problem in the image.

Given Details:

1. The functions $f(x) = 2\cos x$ and $g(x) = -\cos x$ are graphed for $x \in [0^\circ, 360^\circ]$.
2. Questions to answer:
   1. Amplitude of $f(x)$.
   2. Period of $g(x)$.
   3. Length of segment $EF$ if $OE = 315^\circ$.
   4. $x$ values for:
      1. $f(x) \geq g(x)$,
      2. $f(x) = 2$,
      3. $f(x)$ is increasing.

Solution:

1. Amplitude of $f(x)$:

The amplitude of a trigonometric function $A\cos x$ is the coefficient of $\cos x$, i.e., $|A|$.

• For $f(x) = 2\cos x$, the amplitude is: $\text{Amplitude} = |2| = 2.$

2. Period of $g(x)$:

The period of $A\cos(Bx)$ or $A\sin(Bx)$ is given by $\frac{360^\circ}{|B|}$.

• For $g(x) = -\cos x$, $B = 1$, so the period is: $\text{Period} = \frac{360^\circ}{1} = 360^\circ.$

3. Length of segment $EF$ for $OE = 315^\circ$:

• At $x = 315^\circ$, calculate $f(315^\circ)$ and $g(315^\circ)$:
  • $f(315^\circ) = 2\cos(315^\circ) = 2\left(\frac{\sqrt{2}}{2}\right) = \sqrt{2}$.
  • $g(315^\circ) = -\cos(315^\circ) = -\left(\frac{\sqrt{2}}{2}\right) = -\frac{\sqrt{2}}{2}$.
• The vertical distance $EF = f(315^\circ) - g(315^\circ)$: $EF = \sqrt{2} - \left(-\frac{\sqrt{2}}{2}\right) = \sqrt{2} + \frac{\sqrt{2}}{2} = \frac{2\sqrt{2}}{2} + \frac{\sqrt{2}}{2} = \frac{3\sqrt{2}}{2}.$

4. Graphical solutions for $x$:

4.4.1. $f(x) \geq g(x)$:

• $f(x) = 2\cos x$, $g(x) = -\cos x$.
• Solve $2\cos x \geq -\cos x$: $3\cos x \geq 0 \quad \Rightarrow \quad \cos x \geq 0.$
• $\cos x \geq 0$ in $[0^\circ, 360^\circ]$ occurs in: [ x \in [0^\circ, 90^\circ] \cup [270^\circ, 360^\circ]. ]

4.4.2. $f(x) = 2$:

• Solve $2\cos x = 2$: $\cos x = 1.$
• $\cos x = 1$ at: $x = 0^\circ.$

4.4.3. $f(x)$ is increasing:

• For $f(x) = 2\cos x$, the derivative $f'(x) = -2\sin x$.
• $f(x)$ is increasing when $f'(x) > 0$: $-2\sin x > 0 \quad \Rightarrow \quad \sin x < 0.$
• $\sin x < 0$ in $[0^\circ, 360^\circ]$ occurs in: $x \in [180^\circ, 360^\circ].$

Final Answers:

1. Amplitude of $f(x)$: $2$.
2. Period of $g(x)$: $360^\circ$.
3. Length of $EF$ for $OE = 315^\circ$: $\frac{3\sqrt{2}}{2}$.
4. $x$-values:
   1. $f(x) \geq g(x)$: $[0^\circ, 90^\circ] \cup [270^\circ, 360^\circ]$.
   2. $f(x) = 2$: $0^\circ$.
   3. $f(x)$ is increasing: $[180^\circ, 360^\circ]$.

Would you like a detailed explanation for any part? Here are some related questions:

1. What is the formula to find the amplitude of a trigonometric function?
2. How do you determine where a trigonometric function is increasing or decreasing?
3. What are the applications of periodicity in trigonometric graphs?
4. How can one visualize the differences between $f(x)$ and $g(x)$ graphically?
5. What role does the derivative play in identifying increasing or decreasing intervals?

Tip: Always sketch a quick graph for clarity when solving trigonometric equations graphically!