Problem 56 - Algebra for College Students

Question

Graph each of the functions. $$ f(x)=\frac{2}{e^{x}-e^{-x}} $$

Step-by-Step Solution

Answer

The function has a vertical asymptote at \( x = 0 \), horizontal asymptote at \( y = 0 \), and approaches infinity near the vertical asymptote.

Step 1: Understand the function.

The function given to us is \( f(x) = \frac{2}{e^{x} - e^{-x}} \). This function is defined everywhere except where the denominator is zero, as division by zero is undefined.

Step 2: Determine undefined points.

The function is undefined where \( e^x - e^{-x} = 0 \). Solving this equation gives \( e^{2x} = 1 \), yielding \( x = 0 \). This means the function has a vertical asymptote at \( x = 0 \).

Step 3: Analyze the function as \( x \to \infty \) and \( x \to -\infty \).

As \( x \to \infty \), \( e^x \) dominates, so \( e^x - e^{-x} \approx e^x \). Thus, \( f(x) \approx \frac{2}{e^x} \to 0 \). As \( x \to -\infty \), \( e^{-x} \) dominates, similarly leading to \( f(x) \to 0 \).

Step 4: Determine the y-intercept.

The function does not have a y-intercept since it is undefined at \( x = 0 \), which is where the y-axis is located.

Step 5: Determine behavior near the vertical asymptote.

As \( x \to 0^+ \) (approaching from the right), \( e^x - e^{-x} \to 0^+ \), so \( f(x) \to \infty \). As \( x \to 0^- \), \( e^x - e^{-x} \to 0^- \), so \( f(x) \to -\infty \).

Step 6: Sketch the graph.

Draw a vertical line at \( x = 0 \) to represent the asymptote. For very large or very small values of \( x \), draw the curve approaching the \( x \)-axis (horizontal asymptote at \( y = 0 \)). The graph will approach \( \infty \) as \( x \to 0^+ \) and \( -\infty \) as \( x \to 0^- \).

Step 7: Verify and refine the sketch by plotting points.

You can plot specific points for further accuracy, such as \( f(1) \) and \( f(-1) \), and refine the sketch to ensure it approaches infinity as predicted near the asymptotes.

Key Concepts

Vertical Asymptote

Vertical asymptotes occur in rational functions when the denominator equals zero, making the function undefined at that point. They represent values of the input variable, often denoted by \( x \), where the function tends to \( \infty \) or \( -\infty \). For the function \( f(x) = \frac{2}{e^x - e^{-x}} \), the denominator is zero when \( e^x - e^{-x} = 0 \). Solving this equation, we find that \( x = 0 \) is a critical point. Thus, there is a vertical asymptote at \( x = 0 \). This line represents a boundary that the graph approaches but never touches. Understanding vertical asymptotes assists in predicting the behavior of the function as it nears those critical points.

Horizontal Asymptote

Horizontal asymptotes are lines that describe the behavior of a function as \( x \) moves towards positive or negative infinity. These asymptotes help confirm where the graph of a function levels off, serving as a boundary that the function approaches but does not necessarily reach. For \( f(x) = \frac{2}{e^x - e^{-x}} \), we look at the function as \( x \to \infty \) and \( x \to -\infty \). In both cases, the function trends toward zero since the exponential terms \( e^x \) and \( e^{-x} \) grow disproportionately large compared to the numerator. Therefore, the horizontal asymptote of this function is \( y = 0 \), indicating that as \( x \) goes to infinity or negative infinity, \( f(x) \) approaches zero.

Undefined Points

Undefined points in a function occur when an operation within the function is not mathematically valid, such as division by zero. Identifying these points is crucial for graphing and understanding the behavior of the function. In \( f(x) = \frac{2}{e^x - e^{-x}} \), the function is undefined when \( e^x - e^{-x} = 0 \) because division by zero is not permitted. This equation resolves to give \( x = 0 \), which is the undefined point in this function. At this value, the function does not produce a valid output and signifies the presence of a vertical asymptote, where the behavior around this point requires closer examination.

Behavior of Functions at Infinity

The behavior of a function as \( x \) approaches positive or negative infinity is vital in plotting and understanding its long-term trends. This behavior is typically described by the function's horizontal asymptotes. For \( f(x) = \frac{2}{e^x - e^{-x}} \), as \( x \) approaches infinity, the term \( e^x \) grows significantly larger than \( e^{-x} \), making the denominator approximately \( e^x \). Consequently, \( f(x) \) approximates to \( \frac{2}{e^x} \), which simplifies to zero. Similarly, as \( x \to -\infty \), \( e^{-x} \) outweighs \( e^x \), also driving the function towards zero. Thus, in both directions toward infinity, the behavior confirms that \( f(x) \) approaches zero, aligning with the identified horizontal asymptote.