For each of the following, find the critical properties necessary and sketch the function. Describe each special case and state the domain of the function: f(x) = (x-2)/(x+1).

Steps to Solve

1. Identify the function and domain The function given is ( f(x) = \frac{x - 2}{x + 1} ). The domain consists of all real numbers except where the denominator is zero. This gives us: $$ x + 1 \neq 0 \implies x \neq -1 $$ Thus, the domain is ( (-\infty, -1) \cup (-1, \infty) ).

2. Find vertical asymptotes Vertical asymptotes occur where the function is undefined. Here, we set ( x + 1 = 0 ): $$ x = -1 $$ This means there is a vertical asymptote at ( x = -1 ).

3. Find horizontal asymptotes To find horizontal asymptotes, we consider the behavior of ( f(x) ) as ( x ) approaches infinity. We analyze: $$ \lim_{x \to \infty} f(x) \text{ and } \lim_{x \to -\infty} f(x) $$ Both limits simplify to: $$ \lim_{x \to \infty} \frac{x - 2}{x + 1} = 1 \text{ and } \lim_{x \to -\infty} \frac{x - 2}{x + 1} = 1 $$ Thus, the horizontal asymptote is at ( y = 1 ).

4. Identify intercepts

• X-intercept: Set ( f(x) = 0 ): $$ \frac{x-2}{x+1} = 0 \implies x - 2 = 0 \implies x = 2 $$ So the x-intercept is at ( (2, 0) ).

• Y-intercept: Evaluate ( f(0) ): $$ f(0) = \frac{0 - 2}{0 + 1} = -2 $$ Thus, the y-intercept is at ( (0, -2) ).

5. Sketch the graph Using the information above, we can sketch the graph:

• Vertical asymptote at ( x = -1 )
• Horizontal asymptote at ( y = 1 )
• X-intercept at ( (2, 0) )
• Y-intercept at ( (0, -2) )