Question

which graph represents the function $f(x)=\frac{3x - 2}{x - 2}$?

Explanation:

Step1: Find the vertical asymptote

For a rational function \(y = \frac{f(x)}{g(x)}\), the vertical asymptote occurs when \(g(x)=0\).
Set \(x - 2=0\), then \(x = 2\).

Step2: Analyze the horizontal asymptote

For \(y=\frac{3x - 2}{x - 2}\), since the degree of the numerator and the denominator is the same (both degree \(1\)).
The horizontal asymptote is \(y=\frac{3}{1}=3\) (using the formula \(y=\frac{a_n}{b_n}\) for \(y=\frac{a_nx^n+\cdots}{b_nx^n+\cdots}\)).

Step3: Check the \(y\) - intercept

Set \(x = 0\), then \(y=\frac{3\times0 - 2}{0 - 2}=\frac{- 2}{-2}=1\).

Step4: Check the \(x\) - intercept

Set \(y = 0\), then \(3x-2=0\), so \(x=\frac{2}{3}\).

Answer:

The graph with a vertical asymptote at \(x = 2\), a horizontal asymptote at \(y = 3\), \(x\) - intercept at \((\frac{2}{3},0)\) and \(y\) - intercept at \((0,1)\) is the correct graph of the function \(y=\frac{3x - 2}{x - 2}\).