Problem 106 - Precalculus

Question

Use a graphing utility to graph \(y=\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}} .\) State the domain. Determine whether there are any symmetry and asymptote.

Step-by-Step Solution

Answer

Domain: all real numbers except 0; symmetry: origin; asymptotes: vertical at \(x = 0\), horizontal at \(y = 1\) and \(y = -1\).

Step 1: Examine the Domain

To find the domain of the function, we need to identify where the denominator does not equal zero. The denominator is given by \[e^{x} - e^{-x}\] which can be rewritten as \[e^{x} - \frac{1}{e^{x}} = 0\]. This simplifies to \[e^{2x} = 1\]. Solving for \(x\), we find that \(x = 0\). Thus, the domain is all real numbers except \(x = 0\).

Step 2: Identify Symmetry

We'll check if the function is even or odd, indicating symmetry about the y-axis or origin, respectively. A function is even if \(f(x) = f(-x)\) and odd if \(f(x) = -f(-x)\). \[\frac{e^{-x} + e^{x}}{e^{-x} - e^{x}} = \frac{e^{x} + e^{-x}}{-(e^{x} - e^{-x})} = -\frac{e^{x} + e^{-x}}{e^{x} - e^{-x}}\]Since \(y(x) = -y(-x)\), the function is odd and has origin symmetry.

Step 3: Find Asymptotes

Vertical asymptotes occur where the denominator is zero; we've found this occurs at \(x = 0\). Check for horizontal asymptotes by evaluating the limits as \(x\) approaches infinity:\[\lim_{{x \to \infty}} \frac{e^{x} + e^{-x}}{e^{x} - e^{-x}} = 1\] and similarly, \[\lim_{{x \to -\infty}} \frac{e^{x} + e^{-x}}{e^{x} - e^{-x}} = -1.\]Thus, the horizontal asymptotes are \(y = 1\) and \(y = -1\).

Step 4: Graphing the Function

Use a graphing utility to plot the function based on the information gathered from the previous steps. Make sure to include the vertical asymptote at \(x = 0\) and horizontal asymptotes at \(y = 1\) and \(y = -1\). The graph should reflect the origin symmetry determined earlier.

Key Concepts

Domain of a Function

In mathematics, the domain of a function represents all the possible input values (typically denoted as "x") that the function can accept. To determine the domain of a function, you need to ensure that calculations such as division, square roots, and logarithms are defined and valid.

For the function we are examining, which is given by \[y = \frac{e^x + e^{-x}}{e^x - e^{-x}},\]the domain concerns identifying when the denominator equals zero, since division by zero is undefined.

Firstly, let’s analyze the denominator:\[e^x - e^{-x} = 0.\]Rearranging gives:\[e^{2x} = 1.\]Solving for \(x\), you'll find:\[x = 0.\]This solution implies that the function is undefined at \(x = 0\).

Thus, the domain of the function includes all real numbers except \(x = 0\). This is expressed in interval notation as \(-\infty, 0\) \cup (0, \infty).

Symmetry of Functions

Understanding the symmetry of a function can greatly simplify its analysis and graphing. There are two common types of symmetry in functions: even and odd.

- A function is **even** if its graph is symmetrical with respect to the y-axis. Mathematically, this means \[f(x) = f(-x).\]- A function is **odd** if its graph is symmetrical with respect to the origin. This is described by \[f(x) = -f(-x).\]For the function \[y = \frac{e^x + e^{-x}}{e^x - e^{-x}},\]let us check if it is even or odd:

The expression \[\frac{e^{-x} + e^x}{e^{-x} - e^x}\]can be rewritten as:\[-\frac{e^x + e^{-x}}{e^x - e^{-x}}.\]This comparison shows:\[y(-x) = -y(x).\]Thus, the function is odd and has symmetry around the origin. This means reflecting over both the x-axis and y-axis results in the original shape of the function.

Asymptotes

Asymptotes are lines that the graph of a function approach but never touch or cross. There are mainly two types: vertical and horizontal asymptotes.

- **Vertical Asymptotes** occur where the function's denominator is equal to zero and the function itself is undefined. In our function \[y = \frac{e^x + e^{-x}}{e^x - e^{-x}},\]the denominator \[e^x - e^{-x} = 0\] at \(x = 0\) creates a vertical asymptote there.

- **Horizontal Asymptotes**, on the other hand, describe the behavior of a function as \(x\) approaches infinity or negative infinity. To find horizontal asymptotes, calculate the limits:\[\lim_{{x \to \infty}} \frac{e^x + e^{-x}}{e^x - e^{-x}} = 1\]and\[\lim_{{x \to -\infty}} \frac{e^x + e^{-x}}{e^x - e^{-x}} = -1.\]Hence, the horizontal asymptotes are \(y = 1\) and \(y = -1\).

Identifying asymptotes is crucial for understanding the general behavior and direction of a function as it tails off towards infinity or at its domain boundaries.