Find the domain of `f(x) = (x^(2) + 3x + 5)/(x^(2) - 5x+ 4)`

Answer

To find the domain of the function \( f(x) = \frac{x^2 + 3x + 5}{x^2 - 5x + 4} \), we need to determine the values of \( x \) for which the function is defined. A rational function is defined everywhere except where its denominator is equal to zero. Therefore, we need to find the values of \( x \) that make the denominator zero.

### Step 1: Set the denominator equal to zero
We start by setting the denominator equal to zero:
\[ x^2 - 5x + 4 = 0 \]

### Step 2: Factor the quadratic equation
Next, we need to factor the quadratic equation. We look for two numbers that multiply to \( 4 \) (the constant term) and add to \( -5 \) (the coefficient of \( x \)):
\[ (x - 4)(x - 1) = 0 \]

### Step 3: Solve for \( x \)
Now we can solve for \( x \) by setting each factor equal to zero:
1. \( x - 4 = 0 \) → \( x = 4 \)
2. \( x - 1 = 0 \) → \( x = 1 \)

### Step 4: Determine the domain
The function \( f(x) \) is undefined at \( x = 4 \) and \( x = 1 \). Therefore, the domain of \( f(x) \) includes all real numbers except these two values. We can express the domain in interval notation:
\[ \text{Domain of } f(x) = (-\infty, 1) \cup (1, 4) \cup (4, \infty) \]

### Final Answer
Thus, the domain of the function \( f(x) \) is:
\[ \text{Domain} = \mathbb{R} \setminus \{1, 4\} \]

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