2. sketch the graph of the following function. clearly identify all intercepts and asymptotes.\n\n$$f(x) =…

Explanation:

Step1: Find the y-intercept

Set $x = 0$ in the function $f(x) = \frac{6 - 2x}{1 - x}$. $$f(0) = \frac{6 - 2(0)}{1 - 0} = \frac{6}{1} = 6$$

Step2: Find the x-intercept

Set $f(x) = 0$ and solve for $x$. $$6 - 2x = 0 \implies 2x = 6 \implies x = 3$$

Step3: Find the vertical asymptote

Identify where the denominator equals zero. $$1 - x = 0 \implies x = 1$$

Step4: Find the horizontal asymptote

Compare the degrees of the numerator and denominator. $$y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}} = \frac{-2}{-1} = 2$$

Step5: Determine behavior near asymptotes

Analyze the sign of $f(x)$ as $x \to 1^{-}$ and $x \to 1^{+}$. As $x \to 1^{-}$, $f(x) \to \frac{4}{0^{+}} = +\infty$. As $x \to 1^{+}$, $f(x) \to \frac{4}{0^{-}} = -\infty$.

Answer:

y-intercept: $(0, 6)$ x-intercept: $(3, 0)$ Vertical Asymptote: $x = 1$ Horizontal Asymptote: $y = 2$ Graph description: The graph consists of two branches. The left branch passes through $(0, 6)$, approaching $y=2$ as $x \to -\infty$ and $x=1$ as $y \to \infty$. The right branch passes through $(3, 0)$, approaching $y=2$ as $x \to \infty$ and $x=1$ as $y \to -\infty$.