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). $\text{ƒ}\left(x\right)=1/(x-2)$

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1. Identify the vertical asymptote by setting the denominator equal to zero: solve \(x - 2 = 0\) to find \(x = 2\). This means the graph has a vertical asymptote at \(x = 2\).
2. Analyze the behavior of the function \(ƒ(x) = \frac{1}{x-2}\) as \(x\) approaches 2 from the left side (\(x \to 2^-\)). Since the denominator approaches zero from the negative side, the function values will tend toward negative or positive infinity depending on the sign.
3. Analyze the behavior of the function as \(x\) approaches 2 from the right side (\(x \to 2^+\)). Since the denominator approaches zero from the positive side, the function values will tend toward positive or negative infinity accordingly.
4. Compare the behavior near the vertical asymptote with the four given graph choices (A–D). Look for which graph shows the function going to opposite infinities on either side of \(x=2\), consistent with \(\frac{1}{x-2}\).
5. Confirm that the function does not cross the vertical asymptote and that the graph matches the expected shape of \(ƒ(x) = \frac{1}{x-2}\), which typically has two branches, one in the second quadrant and one in the fourth quadrant relative to \(x=2\).

Key Concepts

Vertical Asymptotes

A vertical asymptote occurs in the graph of a function where the function approaches infinity or negative infinity as the input approaches a specific value. For rational functions, vertical asymptotes happen at values of x that make the denominator zero, provided the numerator is not zero at those points.

Behavior Near Vertical Asymptotes

The graph of a rational function near a vertical asymptote can approach positive or negative infinity from either side. Understanding whether the function values go to +∞ or -∞ on each side helps identify the correct graph and distinguish between different rational functions with the same vertical asymptote.

Rational Functions and Their Graphs

A rational function is the ratio of two polynomials. Its graph can have vertical asymptotes, horizontal or oblique asymptotes, and intercepts. Analyzing the function's formula, especially the denominator and numerator, helps predict the shape and key features of its graph.