Problem 33 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the function. Determine its domain and identify any vertical or horizontal asymptotes. $$f(x)=\frac{2+x}{1-x}$$

Step-by-Step Solution

Answer

The graph of the function \(f(x)=\frac{2+x}{1-x}\) has a domain of all real numbers except 1. The function has a vertical asymptote at \(x = 1\) and a horizontal asymptote at \(y = -1\).

Step 1: Identify the Domain

The domain of a function is the set of all valid input values (x-values). For this function, is all real numbers except for those that would make the denominator zero, as division by zero is undefined. Solving the equation \(1 - x = 0\) gives \(x = 1\). Therefore, the domain of the function is \(x \in R\) such that \(x ≠ 1\).

Step 2: Identify the Vertical Asymptotes

Vertical asymptotes are found where the denominator of a fraction is zero but the numerator is not. Here, setting the denominator equal to zero, we get \(1 - x = 0\), which gives us \(x = 1\). Thus, the function has a vertical asymptote at \(x = 1\).

Step 3: Identify the Horizontal Asymptotes

Horizontal asymptotes reflect the end behavior of the graph. They are found by examining the degrees of the numerator and denominator. If the degree of the numerator is less than the degree of the denominator, the x-axis (y = 0) is the horizontal asymptote. If the degrees are the same, the horizontal asymptote is the ratio of the leading coefficients. If the degree of the numerator is greater than the denominator, there is no horizontal asymptote. Here, the degrees of the numerator and denominator are the same (1), and the coefficient of the highest degree term in the numerator is 1 and the denominator is -1. So the horizontal asymptote is \(y = -1\).

Step 4: Graph the Function

To get a visual representation of the function and its asymptotes, plot the function \(f(x)=\frac{2+x}{1-x}\) using a graphing utility. The graph shows the function approaching but never crossing the lines \(x = 1\) (the vertical asymptote) and \(y = -1\) (the horizontal asymptote).

Key Concepts

Domain of a Function

Understanding the domain of a function is crucial when graphing rational functions. For instance, consider the function \(f(x)=\frac{2+x}{1-x}\). To determine its domain, we need to find all possible x-values that won't cause any mathematical errors, like division by zero. Since the denominator \(1-x\) becomes zero when \(x=1\), we exclude this value from the domain. Thus, the domain consists of all real numbers except for \(x=1\). It's often helpful to express this in interval notation: \( (-\infty, 1) \cup (1, \infty) \). Understanding domains helps us to anticipate the limits within which the function can operate and become aware of values that we should avoid when using a graphing utility or plotting points manually.

Vertical Asymptotes

Vertical asymptotes are like the boundaries that a function's graph cannot cross. They occur at values where the function is undefined—typically where the denominator of a rational function is zero. For the given function \(f(x)=\frac{2+x}{1-x}\), the denominator is zero when \(x=1\), thus this is where we find a vertical asymptote. At \(x=1\), the function's value shoots up to infinity, which we represent graphically with a dashed line. The function will approach this line, but no valid y-value can ever be calculated at this point. The concept of vertical asymptotes helps students understand the behavior of graphs near points of undefined values.

Horizontal Asymptotes

The long-term behavior of a function is framed by horizontal asymptotes. How a function behaves as \(x\) moves towards \(\pm\infty\) is what's at play here. When the degrees of the numerator and denominator of a rational function are equal, the horizontal asymptote is determined by the ratio of their leading coefficients. In the function \(f(x)=\frac{2+x}{1-x}\), the numerator and denominator both have a degree of 1, so we compare the coefficients. The ratio is \(\frac{1}{-1}\), which gives us a horizontal asymptote at \(y=-1\). This means as \(x\) stretches toward \(\infty\) or \( -\infty\), the \(y\) values get closer and closer to \( -1\), never surpassing this boundary.

Graphing Utility

Graphing utilities, such as online graphing calculators or software, are invaluable tools when learning about functions. They simplify the process of visualizing complex functions like \(f(x)=\frac{2+x}{1-x}\). By inputting the function's formula, we can quickly see its shape, along with features like intercepts, asymptotes, and the general trend of the graph. When graphing rational functions, a graphing utility not only illustrates the domain and range of the function but also how it behaves near asymptotes—it never touches vertical asymptotes and gets infinitely closer to horizontal ones. This provides a powerful visual aid to students who are grappling with abstract concepts and enhances their understanding of the lecture material or textbook solutions by bringing these concepts to life.