Find the range of each of the following function:
` f(x)=cos^(-1)(((x-1)(x+5))/(x(x-2)(x-3)))`

Explanation

To find the range of the function \( f(x) = \cos^{-1}\left(\frac{(x-1)(x+5)}{x(x-2)(x-3)}\right) \), we need to determine the values that the expression inside the inverse cosine function can take.

### Step 1: Determine the domain of the function inside the inverse cosine

The function \( \cos^{-1}(t) \) is defined for \( t \) in the range \([-1, 1]\). Therefore, we need to find the values of \( x \) such that:

\[ -1 \leq \frac{(x-1)(x+5)}{x(x-2)(x-3)} \leq 1 \]

### Step 2: Solve the inequalities

We will split this into two inequalities:

1. **First Inequality**:

\[ \frac{(x-1)(x+5)}{x(x-2)(x-3)} \leq 1 \]

Rearranging gives:

\[ (x-1)(x+5) \leq x(x-2)(x-3) \]

This simplifies to:

\[ (x-1)(x+5) - x(x-2)(x-3) \leq 0 \]

Expanding both sides and combining like terms will help us find the critical points.

2. **Second Inequality**:

\[ \frac{(x-1)(x+5)}{x(x-2)(x-3)} \geq -1 \]

Rearranging gives:

\[ (x-1)(x+5) + x(x-2)(x-3) \geq 0 \]

Again, expand and simplify to find the critical points.

### Step 3: Find critical points

After solving both inequalities, we will find the critical points where the function changes sign. This will involve finding the roots of the polynomial expressions obtained from the inequalities.

### Step 4: Test intervals

Using the critical points, we will test intervals to determine where the inequalities hold true. This will help us find the valid \( x \) values that keep the expression within the bounds of \([-1, 1]\).

### Step 5: Determine the range of \( f(x) \)

Once we have the valid \( x \) values, we can evaluate the corresponding \( f(x) \) values. The range of \( f(x) \) will be determined by the values of \( \cos^{-1}(t) \) for \( t \) in the interval \([-1, 1]\).

Since \( \cos^{-1}(t) \) takes values from \( [0, \pi] \), the range of \( f(x) \) will be:

\[ \text{Range of } f(x) = [0, \pi] \]

### Final Answer:

The range of the function \( f(x) = \cos^{-1}\left(\frac{(x-1)(x+5)}{x(x-2)(x-3)}\right) \) is \([0, \pi]\).

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