identify the rational function whose graph is given below.

Explanation:

Step1: Identify vertical asymptote

The vertical asymptote is at (x = 3). For a rational function (y=\frac{f(x)}{g(x)}), the vertical asymptote occurs at the values of (x) that make (g(x)=0). So the denominator has a factor of ((x - 3)).

Step2: Identify horizontal asymptote

The horizontal asymptote is (y = 2). For a rational function (\frac{ax^{n}}{bx^{m}}) where (n=m), the horizontal asymptote is (y=\frac{a}{b}). Let the rational - function be (y=\frac{2x + k}{x - 3}) (since the horizontal asymptote is (y = 2)).

Step3: Find the value of (k)

We can use a point on the graph. Let's assume the graph passes through the point ((0,2)) (by looking at the (y) - intercept). Substitute (x = 0) and (y=2) into (y=\frac{2x + k}{x - 3}). We get (2=\frac{2\times0 + k}{0 - 3}), which simplifies to (2=\frac{k}{-3}), and then (k=-6). So the rational function is (y=\frac{2x-6}{x - 3}).

Answer:

(y=\frac{2x - 6}{x - 3})