Problem 96 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

The function \(g(x)=\frac{x^{2}+x-2}{x-1}\) simplifies to \(g(x)=x+2\), which is a straight linear function. Although from the first glance it might seem that this function would have a vertical asymptote at \(x = 1\), the fact that \(x-1\) factor cancels out in the numerator and denominator in the simplified form of the function reveals that there are no vertical asymptotes.

Step 1: Simplify the function

The function \(g(x)=\frac{x^{2}+x-2}{x-1}\) can be simplified to \(g(x)= x+1\). This simplification is performed by factoring the numerator, which yields \(x^{2}+x-2=(x-1)(x+2)\). This simplifies the function to \(\frac{(x-1)(x+2)}{x-1}\), and allows the cancellation of \(x-1\) from the numerator and denominator, resulting in \(g(x)=x+2\).

Step 2: Graph the Function

The graph of the function \(g(x)=x+2\) is a straight line with a slope of 1 and a y-intercept of 2. The function continues indefinitely in the positive and negative direction of both the x and y axis.

Step 3: Identify the Absence of Vertical Asymptotes

Usually, vertical asymptotes occur when the denominator of a fraction equals zero, because it causes the function to be undefined. In the original expression of \(g(x)=\frac{x^{2}+x-2}{x-1}\), the denominator would be zero when \(x=1\). However, the simplification cancels out the term in the denominator that would have caused the function to be undefined at \(x=1\), therefore there are no vertical asymptotes.

Key Concepts

Understanding Vertical Asymptotes

Vertical asymptotes are straight lines that a graph approaches but never touches or crosses. They occur in a graph of a rational function when a value of x makes the denominator equal to zero while the numerator is non-zero, leading to an undefined condition. In mathematical terms, if you have a rational function like \( f(x) = \frac{N(x)}{D(x)} \), and \( D(x_0) = 0 \) while \( N(x_0) eq 0 \), then \( x = x_0 \) would be a vertical asymptote.

In the given exercise, a vertical asymptote was expected at \( x=1 \) because substituting \( x=1 \) into the denominator of \( g(x) \) would yield zero. However, upon factoring and simplifying, the term causing the zero in the denominator (\( x-1 \)) was canceled out, resulting in a linear equation rather than a rational function. Therefore, in this case, the graph does not display a vertical asymptote at \( x = 1 \) since the function is not undefined there. It's crucial to simplify rational expressions when looking for vertical asymptotes, as they might not exist post simplification.

Simplifying Rational Expressions

Simplifying rational expressions is a critical skill in algebra that can reveal the true nature of a function. It generally involves factoring polynomials in the numerator and denominator and eliminating common factors. Simplification can alter the graph's characteristics, like removing or adding vertical asymptotes.

To simplify a rational expression like \( g(x)=\frac{x^{2}+x-2}{x-1} \), we look for common factors in the numerator and denominator. Factoring the numerator gives \( x^{2}+x-2 = (x+2)(x-1) \). Since the \( x-1 \) term is present in both numerator and denominator, they cancel each other. After canceling, we end up with \( g(x) = x+2 \), which is a linear equation, not a rational expression. Always check for factors that can be canceled to simplify your rational functions; this will not only make them easier to work with, but also elucidate their graphical features.

Graphing Linear Equations

Linear equations form the foundation of algebra and their graphs represent straight lines on the coordinate plane. The general form of a linear equation is \( y = mx + b \), where \( m \) is the slope and \( b \) represents the y-intercept. The slope tells you how steep the line is, and the y-intercept is where the line crosses the y-axis.

When graphing the simplified function \( g(x)=x+2 \), you can see that it is a linear equation with a slope of 1 and a y-intercept of 2. Consequently, it produces a straight line on the graph that increases by 1 unit vertically for every 1 unit increased horizontally. This graph extends indefinitely in both x and y directions and has no vertical asymptote since it's a continuous line. If graphing by hand, you would plot the y-intercept at (0,2), then use the slope to find another point by moving up 1 unit and 1 unit to the right from the y-intercept. Draw a line through these two points, and you have graphed the equation. Graphing is a valuable tool for visualizing equations, making it easier to understand and interpret their behaviors and solutions.