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

Options

A. R
B. R - { 1,4}
C. R - {1}
D. (1,4)

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. The function is undefined when the denominator is equal to zero. Therefore, we need to solve the equation:

1. **Set the denominator equal to zero:**
\[ x^2 - 5x + 4 = 0 \]

2. **Factor the quadratic equation:**
To factor \( x^2 - 5x + 4 \), we look for two numbers that multiply to \( 4 \) (the constant term) and add up to \( -5 \) (the coefficient of \( x \)). The numbers \( -1 \) and \( -4 \) work:
\[ (x - 1)(x - 4) = 0 \]

3. **Find the roots:**
Set each factor equal to zero:
\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]
\[ x - 4 = 0 \quad \Rightarrow \quad x = 4 \]

4. **Determine the values to exclude from the domain:**
The function is undefined at \( x = 1 \) and \( x = 4 \). Therefore, these values must be excluded from the domain.

5. **Write the domain in interval notation:**
The domain of \( f(x) \) consists of all real numbers except \( 1 \) and \( 4 \). In interval notation, this can be expressed as:
\[ \text{Domain of } f = (-\infty, 1) \cup (1, 4) \cup (4, \infty) \]

Thus, the domain of the function \( f(x) = \frac{x^2 + 3x + 5}{x^2 - 5x + 4} \) is:
\[ (-\infty, 1) \cup (1, 4) \cup (4, \infty) \]