Work each problem. Choices A–D below show the four ways in which the graph of a rational function can approach the vertical line x=2 as an asymptote. Identify the graph of each rational function defined in parts (a) – (d). $\text{ƒ}\left(x\right)=-1/(x-2)$

Verified step by step guidance

1. Step 1: Identify the vertical asymptote of the function. The function is given as \(ƒ(x) = -\frac{1}{x-2}\), so the vertical asymptote occurs where the denominator is zero, which is at \(x=2\).
2. Step 2: Analyze the behavior of the function near the vertical asymptote \(x=2\). Since the numerator is negative (-1), the function will approach positive or negative infinity depending on the side from which \(x\) approaches 2.
3. Step 3: Consider the limit as \(x\) approaches 2 from the left (\(x \to 2^-\)). The denominator \((x-2)\) is negative and close to zero, so \(-\frac{1}{x-2}\) becomes a large positive number (because dividing by a small negative number gives a large negative, but the negative sign in front flips it to positive). Thus, the graph goes to \(+\infty\) on the left side of \(x=2\).
4. Step 4: Consider the limit as \(x\) approaches 2 from the right (\(x \to 2^+\)). The denominator \((x-2)\) is positive and close to zero, so \(-\frac{1}{x-2}\) becomes a large negative number. Thus, the graph goes to \(-\infty\) on the right side of \(x=2\).
5. Step 5: Match this behavior to the graphs provided. The graph should show the function going to positive infinity on the left side of \(x=2\) and negative infinity on the right side of \(x=2\). This corresponds to graph A.

Key Concepts

Vertical Asymptotes of Rational Functions

A vertical asymptote occurs where the denominator of a rational function is zero, causing the function to approach infinity or negative infinity. For the function f(x) = -1/(x-2), the vertical asymptote is at x = 2, where the function is undefined and the graph approaches a vertical line.

Behavior Near Vertical Asymptotes

The behavior of a rational function near a vertical asymptote depends on the sign of the function on either side of the asymptote. For f(x) = -1/(x-2), as x approaches 2 from the left, the function tends to positive infinity, and from the right, it tends to negative infinity, indicating opposite behavior on each side.

Graphing Rational Functions

Graphing rational functions involves identifying asymptotes and understanding the function's behavior near these lines. The sign and magnitude of the function near the vertical asymptote help determine which graph matches the function, as seen in the choices where the graph approaches infinity or negative infinity differently on each side of x=2.