The domain of definition of the function ` f (x) = sin^(-1) ((x-3)//(2)) - log (4-x)` is

Options

A. ` 1 le x le 5`
B. ` 1 lt x lt 4`
C. ` 1 le x le 4`
D. ` 1le x lt 4`

Answer

To find the domain of the function \( f(x) = \sin^{-1}\left(\frac{x-3}{2}\right) - \log(4-x) \), we need to analyze the conditions for both the inverse sine function and the logarithmic function.

### Step 1: Determine the domain for the inverse sine function

The function \( \sin^{-1}(y) \) is defined for \( y \) in the interval \([-1, 1]\). Here, we have:

\[ y = \frac{x-3}{2} \]

Thus, we need:

\[ -1 \leq \frac{x-3}{2} \leq 1 \]

**Hint:** To solve this inequality, multiply all parts by 2 to eliminate the fraction.

### Step 2: Solve the inequality

Multiplying the entire inequality by 2 gives:

\[ -2 \leq x - 3 \leq 2 \]

Now, we can break this into two parts:

1. \( -2 \leq x - 3 \)
2. \( x - 3 \leq 2 \)

**Hint:** Solve each part separately by isolating \( x \).

### Step 3: Isolate \( x \)

1. For \( -2 \leq x - 3 \):

\[ x \geq 1 \]

2. For \( x - 3 \leq 2 \):

\[ x \leq 5 \]

Combining these results, we get:

\[ 1 \leq x \leq 5 \]

**Hint:** This gives us the interval \( [1, 5] \).

### Step 4: Determine the domain for the logarithmic function

The logarithmic function \( \log(z) \) is defined for \( z > 0 \). Here, we have:

\[ z = 4 - x \]

Thus, we need:

\[ 4 - x > 0 \]

**Hint:** Rearrange this inequality to find the condition for \( x \).

### Step 5: Solve the inequality

Rearranging gives:

\[ x < 4 \]

**Hint:** This tells us that \( x \) must be less than 4.

### Step 6: Combine the results

Now we have two conditions:

1. \( 1 \leq x \leq 5 \)
2. \( x < 4 \)

To find the domain of \( f(x) \), we take the intersection of these two conditions:

The first condition gives us the interval \( [1, 5] \), and the second condition restricts it to \( (-\infty, 4) \).

Thus, the intersection is:

\[ [1, 4) \]

**Final Answer:** The domain of the function \( f(x) \) is \( [1, 4) \).