Draw the graphs of the following function
`abs(f(x))=cosx xin[-2pi,2pi]`

Answer

To draw the graph of the function given by \(|f(x)| = \cos x\) for \(x \in [-2\pi, 2\pi]\), we will follow these steps:

### Step 1: Understand the Function
The equation \(|f(x)| = \cos x\) implies two cases:
1. When \(f(x) \geq 0\), then \(f(x) = \cos x\).
2. When \(f(x) < 0\), then \(f(x) = -\cos x\).

### Step 2: Identify the Domain
We need to plot the graphs for \(x\) in the interval \([-2\pi, 2\pi]\).

### Step 3: Plot the Graph of \(\cos x\)
First, we will plot the graph of \(\cos x\) within the given interval:
- The cosine function oscillates between 1 and -1.
- Key points to note:
- \(\cos(-2\pi) = 1\)
- \(\cos(-\pi) = -1\)
- \(\cos(0) = 1\)
- \(\cos(\pi) = -1\)
- \(\cos(2\pi) = 1\)

The graph of \(\cos x\) will have a wave-like shape, starting from (−2π, 1), going down to (−π, -1), up to (0, 1), down to (π, -1), and finally back to (2π, 1).

### Step 4: Determine the Positive and Negative Parts
From the graph of \(\cos x\):
- The positive part of \(\cos x\) occurs in the intervals:
- \([-2\pi, -\pi/2]\)
- \([-\pi/2, \pi/2]\)
- \([\pi/2, \pi]\)
- \([\pi, 3\pi/2]\)
- \([3\pi/2, 2\pi]\)

- The negative part occurs in the intervals:
- \([-π, -π/2]\)
- \([π/2, π]\)
- \([3\pi/2, 2\pi]\)

### Step 5: Plot \(f(x) = -\cos x\) for Negative Values
For the intervals where \(\cos x\) is negative, we will plot \(f(x) = -\cos x\):
- This will reflect the negative parts of the \(\cos x\) graph across the x-axis.

### Step 6: Combine the Two Graphs
- For the intervals where \(\cos x\) is positive, plot \(f(x) = \cos x\).
- For the intervals where \(\cos x\) is negative, plot \(f(x) = -\cos x\).

### Step 7: Final Graph
The final graph will consist of:
- The segments of \(\cos x\) above the x-axis where \(\cos x\) is positive.
- The segments of \(-\cos x\) below the x-axis where \(\cos x\) is negative.