Find the domain of defination the following functions.
`f(x)=sqrt(3-x)+cos^(-1) ((3-2x)/(5))+ln (2|x|-3)`

Answer

To find the domain of the function \( f(x) = \sqrt{3 - x} + \cos^{-1}\left(\frac{3 - 2x}{5}\right) + \ln(2|x| - 3) \), we need to ensure that each component of the function is defined. Let's analyze each part step by step.

### Step 1: Analyze the square root function

The square root function \( \sqrt{3 - x} \) is defined when the expression inside the square root is non-negative:
\[ 3 - x \geq 0 \]
This implies:
\[ x \leq 3 \]
Thus, the domain for this part is:
\[ (-\infty, 3] \]

### Step 2: Analyze the inverse cosine function

The function \( \cos^{-1}\left(\frac{3 - 2x}{5}\right) \) is defined when the argument lies within the interval \([-1, 1]\):
\[ -1 \leq \frac{3 - 2x}{5} \leq 1 \]
We will break this down into two inequalities.

**First inequality:**
\[ \frac{3 - 2x}{5} \geq -1 \]
Multiplying through by 5:
\[ 3 - 2x \geq -5 \]
Rearranging gives:
\[ 8 \geq 2x \quad \Rightarrow \quad 4 \geq x \quad \Rightarrow \quad x \leq 4 \]

**Second inequality:**
\[ \frac{3 - 2x}{5} \leq 1 \]
Multiplying through by 5:
\[ 3 - 2x \leq 5 \]
Rearranging gives:
\[ -2x \leq 2 \quad \Rightarrow \quad x \geq -1 \]

Combining these two results, we find:
\[ -1 \leq x \leq 4 \]

### Step 3: Analyze the logarithmic function

The function \( \ln(2|x| - 3) \) is defined when its argument is positive:
\[ 2|x| - 3 > 0 \]
This simplifies to:
\[ 2|x| > 3 \quad \Rightarrow \quad |x| > \frac{3}{2} \]
This gives us two cases:
1. \( x > \frac{3}{2} \)
2. \( x < -\frac{3}{2} \)

### Step 4: Combine the domains

Now we need to combine the domains obtained from each part:

1. From the square root: \( (-\infty, 3] \)
2. From the inverse cosine: \( [-1, 4] \)
3. From the logarithm: \( (-\infty, -\frac{3}{2}) \cup (\frac{3}{2}, \infty) \)

**Finding the intersection:**
- The intersection of \( (-\infty, 3] \) and \( [-1, 4] \) is \( [-1, 3] \).
- The intersection of \( [-1, 3] \) and \( (-\infty, -\frac{3}{2}) \cup (\frac{3}{2}, \infty) \) is:
- For \( (-\infty, -\frac{3}{2}) \): The intersection is \( [-1, -\frac{3}{2}) \).
- For \( (\frac{3}{2}, \infty) \): The intersection is \( (\frac{3}{2}, 3] \).

Thus, the overall domain of the function \( f(x) \) is:
\[ [-1, -\frac{3}{2}) \cup \left(\frac{3}{2}, 3\right] \]

### Final Answer:
The domain of the function \( f(x) \) is:
\[ [-1, -\frac{3}{2}) \cup \left(\frac{3}{2}, 3\right] \]