Questions: f(x)=(x-4)/(x-2)

Solution Steps

To analyze the function \( f(x) = \frac{x-4}{x-2} \), we can determine its domain, identify any vertical asymptotes, and find the horizontal asymptote. The domain is all real numbers except where the denominator is zero. Vertical asymptotes occur where the denominator is zero and the numerator is not zero. The horizontal asymptote can be found by examining the behavior of the function as \( x \) approaches infinity.

The function \( f(x) = \frac{x-4}{x-2} \) is defined for all real numbers except where the denominator is zero. Thus, the domain is given by: \[ \text{Domain} = (-\infty, 2) \cup (2, \infty) \]

Vertical asymptotes occur where the denominator is zero and the numerator is not zero. Setting the denominator equal to zero: \[ x - 2 = 0 \implies x = 2 \] Therefore, there is a vertical asymptote at: \[ x = 2 \]

To find the horizontal asymptote, we examine the behavior of \( f(x) \) as \( x \) approaches infinity. The leading coefficients of the numerator and denominator are both 1, so: \[ \lim_{x \to \infty} f(x) = \frac{1}{1} = 1 \] Thus, the horizontal asymptote is: \[ y = 1 \]

Final Answer