Question

Use a graphing utility to graph the function. Explain why there is no vertical asymptote when a superficial examination of the function may indicate that there should be one. \(g(x)=\frac{x^{2}+x-2}{x-1}\)

Short answer

The graph of the function \( g(x) = \frac{x^{2}+x-2}{x-1} \) is a straight line \(g(x) = x + 2\), when \(x\neq1\), and there is no vertical asymptote. This is because the factor causing the function to be undefined at \(x = 1\) gets cancelled out while simplifying the function.

Step-by-step solution

Simplifying the Function

Attempt to simplify the function as it might cancel out the term in denominator. In this case, factorise the numerator \(x^{2} + x - 2\), which can be broken down as \((x - 1) (x + 2)\). Hence, the function simplifies to \(g(x) = x + 2\), when \(x \neq 1\).

Graphing the Simplified Function

The simplified function \(g(x) = x + 2\) represents a straight line with slope 1 and y-intercept 2. Graph this using a graphing utility.

Understanding the Absence of Vertical Asymptote

An apparent vertical asymptote at \(x = 1\) did not appear in the graph, because the term causing this asymptote, \((x - 1)\), cancelled out while simplifying the function. Therefore, although the function is undefined at \(x = 1\), there is no vertical asymptote there.