Find the range of each of the following functions
`cos((x)/(3))`

Answer

To find the range of the function \( f(x) = \cos\left(\frac{x}{3}\right) \), we can follow these steps:

### Step 1: Understand the cosine function
The cosine function, \( \cos(\theta) \), has a well-known range. It oscillates between -1 and 1 for all values of \( \theta \).

### Step 2: Substitute \( \theta \)
Let’s set \( \theta = \frac{x}{3} \). This means that as \( x \) varies over all real numbers, \( \theta \) will also vary over all real numbers.

### Step 3: Analyze the range of \( \cos(\theta) \)
Since \( \theta \) can take any real value, the function \( \cos(\theta) \) will also take all values in its range. Therefore, the range of \( \cos(\theta) \) remains between -1 and 1.

### Step 4: Conclude the range of the original function
Thus, the range of the function \( f(x) = \cos\left(\frac{x}{3}\right) \) is the same as the range of \( \cos(\theta) \), which is:
\[ \text{Range of } f(x) = [-1, 1] \]

### Final Answer:
The range of the function \( f(x) = \cos\left(\frac{x}{3}\right) \) is \([-1, 1]\).

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