Let's break down and solve both parts (a) and (b) of this problem step by step.

(a) Find the range of $f(x) = 1 + 2|\cos(\pi x) + \frac{1}{2}|$, algebraically.

Step 1: Start with the range of $\cos x$

We know that the cosine function, $\cos x$, has a range of $[-1, 1]$.

Step 2: Analyze $\cos(\pi x)$

Since $\cos(\pi x)$ is simply a horizontal stretch of $\cos x$, its range will also be $[-1, 1]$. This is because multiplying the input of the cosine function by $\pi$ does not affect its range.

Step 3: Apply the transformation $\cos(\pi x) + \frac{1}{2}$

The next step is to add $\frac{1}{2}$ to $\cos(\pi x)$. When you add $\frac{1}{2}$ to any value in $[-1, 1]$, the new range becomes: [ \left[ -1 + \frac{1}{2}, 1 + \frac{1}{2} \right] = \left[ -\frac{1}{2}, \frac{3}{2} \right] ]

Step 4: Take the absolute value $|\cos(\pi x) + \frac{1}{2}|$

The absolute value operation affects negative values by making them positive. Thus, the range of $|\cos(\pi x) + \frac{1}{2}|$ becomes: $\left[ 0, \frac{3}{2} \right]$ because the smallest value $-\frac{1}{2}$ is turned into 0.

Step 5: Multiply by 2

Now, multiplying by 2 stretches the range. So, we multiply each value in the range $[0, \frac{3}{2}]$ by 2: $\left[ 0, 3 \right]$

Step 6: Add 1

Finally, adding 1 shifts the entire range upward by 1, so the final range of $f(x)$ becomes: $\left[ 1, 4 \right]$

Thus, the range of $f(x) = 1 + 2|\cos(\pi x) + \frac{1}{2}|$ is $[1, 4]$.

(b) Graph the function $f(x) = 1 + 2|\cos(\pi x) + \frac{1}{2}|$

You are asked to graph this function by starting with the graph of $y = \cos x$ and applying appropriate transformations step by step. Let's outline the transformations:

1. Start with the base graph $y = \cos x$:

   • The cosine graph oscillates between $-1$ and $1$.

2. Apply horizontal scaling to get $\cos(\pi x)$:

   • This stretches the cosine function horizontally by a factor of $\pi$.

3. Shift vertically by $+\frac{1}{2}$ to get $\cos(\pi x) + \frac{1}{2}$:

   • This shifts the graph upward by 0.5 units, so it now oscillates between $-\frac{1}{2}$ and $\frac{3}{2}$.

4. Apply absolute value to get $|\cos(\pi x) + \frac{1}{2}|$:

   • Any negative parts of the graph (those that dipped below 0) are reflected above the x-axis.

5. Multiply by 2 to get $2|\cos(\pi x) + \frac{1}{2}|$:

   • This stretches the graph vertically, so it now oscillates between 0 and 3.

6. Add 1 to get $1 + 2|\cos(\pi x) + \frac{1}{2}|$:

   • Finally, shift the entire graph upward by 1 unit, resulting in the final range from 1 to 4.

These transformations would produce the graph step-by-step. You are asked to draw and annotate each transformation, resulting in six graphs, including the original $y = \cos x$.

Do you want a detailed drawing of each transformation?

Here are five related questions for you:

1. How does the period of $\cos(\pi x)$ compare to $\cos(x)$?
2. What effect does taking the absolute value of a function have on its graph?
3. Why does multiplying by 2 stretch the graph vertically?
4. How do you determine the amplitude of a cosine function after transformations?
5. What happens to the graph of $\cos(x)$ when you shift it vertically by a constant?

Tip: When graphing functions, always apply transformations step by step in the correct order to avoid mistakes!