Let's solve the given questions based on $f(x) = \cos\left(2x + \frac{\pi}{3}\right)$.

(a) What is the range of $f$?

The cosine function, $\cos(x)$, always oscillates between $-1$ and $1$. Since the given function is a standard cosine function (no vertical transformations), the range is: $[-1, 1].$

(b) What is the amplitude of $f$?

The amplitude of a cosine function is the coefficient of the cosine term. In $f(x) = \cos\left(2x + \frac{\pi}{3}\right)$, there is no coefficient other than $1$, so the amplitude is: $1.$

(c) What is the period of $f$?

The general formula for the period of a cosine function is: $\text{Period} = \frac{2\pi}{|b|}$ where $b$ is the coefficient of $x$ in $\cos(bx)$.

Here, $b = 2$. Therefore: $\text{Period} = \frac{2\pi}{2} = \pi.$

(d) By what fraction of the period of $f$ should the graph be shifted left to obtain the graph of $f$?

The phase shift is determined by the term $\frac{\pi}{3}$ in $2x + \frac{\pi}{3}$. The formula for the phase shift is: $\text{Phase shift} = -\frac{c}{b}$ where $c = \frac{\pi}{3}$ and $b = 2$. Substituting: $\text{Phase shift} = -\frac{\frac{\pi}{3}}{2} = -\frac{\pi}{6}.$ This means the graph is shifted left by $\frac{\pi}{6}$.

As a fraction of the period ($\pi$): $\frac{\text{Phase shift}}{\text{Period}} = \frac{\frac{\pi}{6}}{\pi} = \frac{1}{6}.$

Thus, the graph is shifted left by $\frac{1}{6}$ of the period.

(e) Sketch the graph of $f$ on the interval $[-9, 9]$.

The function $f(x) = \cos(2x + \frac{\pi}{3})$ has a period of $\pi$ and is shifted left by $\frac{\pi}{6}$. The graph would be a cosine wave oscillating between $-1$ and $1$, with the phase shift and period taken into account.

I can create a sketch of the graph if you want—let me know!

(f) Sketch the graph of $\cos(2x)$ on the interval $[-9, 9]$.

The function $\cos(2x)$ has the same amplitude and period as $\cos(2x + \frac{\pi}{3})$ but no phase shift. It starts at $1$ at $x = 0$, oscillates between $-1$ and $1$, and repeats every $\pi$.

Again, I can provide the graph if requested.

Let me know if you'd like to see the graphs for parts (e) and (f) or need more clarification!

Related Questions:

1. How do you find the phase shift for a general trigonometric function like $a\cos(bx + c) + d$?
2. What is the effect of changing the amplitude of a cosine function on its graph?
3. How does the coefficient $b$ affect the frequency of the cosine function?
4. What happens to the range of $f(x) = \cos(2x)$ if a vertical shift is applied?
5. How can you use the unit circle to verify the periodicity of trigonometric functions?

Tip:

When graphing trigonometric functions, always start by identifying the amplitude, period, phase shift, and vertical shift. These help in accurately sketching the graph.