Questions: State the domain, intercepts, and asymptotes of the rational function f(x) = (2x-3)/(x-4). If the function does not have the requested characteristic, enter DNE. If there are multiple instances of a characteristic, separate them with commas. 1) Domain (in interval notation): 2) x-intercept(s) (as ordered pair(s)): 3) y-intercept (as ordered pair): 4) Equation(s) of vertical asymptote(s): 5) Equation of horizontal asymptote:

Solution Steps

To solve the given problem, we need to analyze the rational function \( f(x) = \frac{2x - 3}{x - 4} \) and determine its domain, intercepts, and asymptotes.

1. Domain: The domain of a rational function is all real numbers except where the denominator is zero. For \( f(x) \), the denominator \( x - 4 \) is zero when \( x = 4 \). Therefore, the domain is all real numbers except \( x = 4 \).

2. x-intercept(s): The x-intercepts occur where the numerator is zero. Set \( 2x - 3 = 0 \) and solve for \( x \).

3. y-intercept: The y-intercept occurs where \( x = 0 \). Substitute \( x = 0 \) into the function to find \( f(0) \).

4. Vertical asymptote(s): Vertical asymptotes occur where the denominator is zero and the numerator is not zero. For \( f(x) \), this is at \( x = 4 \).

5. Horizontal asymptote: For rational functions where the degrees of the numerator and denominator are the same, the horizontal asymptote is the ratio of the leading coefficients. Here, both the numerator and denominator are linear (degree 1), so the horizontal asymptote is \( y = \frac{2}{1} = 2 \).

The domain of the rational function \( f(x) = \frac{2x - 3}{x - 4} \) is all real numbers except where the denominator is zero. The denominator \( x - 4 \) is zero when \( x = 4 \). Therefore, the domain is: \[ (-\infty, 4) \cup (4, \infty) \]

The \( x \)-intercepts occur where the numerator is zero. Set \( 2x - 3 = 0 \) and solve for \( x \): \[ 2x - 3 = 0 \] \[ x = \frac{3}{2} \] Thus, the \( x \)-intercept is: \[ \left( \frac{3}{2}, 0 \right) \]

The \( y \)-intercept occurs where \( x = 0 \). Substitute \( x = 0 \) into the function: \[ f(0) = \frac{2(0) - 3}{0 - 4} = \frac{-3}{-4} = \frac{3}{4} \] Thus, the \( y \)-intercept is: \[ (0, \frac{3}{4}) \]

Vertical asymptotes occur where the denominator is zero and the numerator is not zero. For \( f(x) \), this is at \( x = 4 \). Therefore, the vertical asymptote is: \[ x = 4 \]

For rational functions where the degrees of the numerator and denominator are the same, the horizontal asymptote is the ratio of the leading coefficients. Here, both the numerator and denominator are linear (degree 1), so the horizontal asymptote is: \[ y = \frac{2}{1} = 2 \]

Final Answer