If f(x) = `sqrt(x ^(2) - 5x + 4)` & g(x) = x + 3 , then find the domain of `(f)/(g) (x)`.

Answer

To find the domain of the function \( \frac{f(x)}{g(x)} \) where \( f(x) = \sqrt{x^2 - 5x + 4} \) and \( g(x) = x + 3 \), we need to ensure that both the numerator and the denominator are defined and that the denominator is not equal to zero.

### Step 1: Determine the conditions for \( f(x) \)

The function \( f(x) \) involves a square root, so we need the expression inside the square root to be non-negative:

\[ x^2 - 5x + 4 \geq 0 \]

### Step 2: Factor the quadratic expression

To solve the inequality, we can factor the quadratic:

\[ x^2 - 5x + 4 = (x - 1)(x - 4) \]

### Step 3: Find the critical points

Setting the factored expression equal to zero gives us the critical points:

\[ (x - 1)(x - 4) = 0 \implies x = 1 \quad \text{and} \quad x = 4 \]

### Step 4: Test intervals around the critical points

We will test the sign of \( (x - 1)(x - 4) \) in the intervals determined by the critical points: \( (-\infty, 1) \), \( (1, 4) \), and \( (4, \infty) \).

1. **Interval \( (-\infty, 1) \)**: Choose \( x = 0 \)
\[ (0 - 1)(0 - 4) = (-1)(-4) = 4 \quad (\text{positive}) \]

2. **Interval \( (1, 4) \)**: Choose \( x = 2 \)
\[ (2 - 1)(2 - 4) = (1)(-2) = -2 \quad (\text{negative}) \]

3. **Interval \( (4, \infty) \)**: Choose \( x = 5 \)
\[ (5 - 1)(5 - 4) = (4)(1) = 4 \quad (\text{positive}) \]

### Step 5: Write the solution for \( f(x) \)

From our tests, \( (x - 1)(x - 4) \geq 0 \) is satisfied in the intervals:

\[ (-\infty, 1] \cup [4, \infty) \]

### Step 6: Determine the conditions for \( g(x) \)

Next, we need to ensure that the denominator \( g(x) = x + 3 \) is not equal to zero:

\[ x + 3 \neq 0 \implies x \neq -3 \]

### Step 7: Combine the conditions

Now we need to combine the intervals from \( f(x) \) and the restriction from \( g(x) \):

1. The intervals from \( f(x) \) are \( (-\infty, 1] \cup [4, \infty) \).
2. We exclude \( x = -3 \) from the domain.

### Step 8: Final domain

The domain of \( \frac{f(x)}{g(x)} \) is:

\[ (-\infty, -3) \cup (-3, 1] \cup [4, \infty) \]

### Summary of the Domain

Thus, the final domain of \( \frac{f(x)}{g(x)} \) is:

\[ (-\infty, -3) \cup (-3, 1] \cup [4, \infty) \]