Problem 45 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

Graph is an S-shaped curve passing through (0, 0), (1, 1.5), (-1, -1.5).

Step 1: Understanding the Function

The given function is \( f(x) = 2^{x} - 2^{-x} \). This is an exponential function involving both \( 2^{x} \) and \( 2^{-x} \), which can be rewritten as \( \frac{1}{2^{x}} \). The graph of \( f(x) \) will involve both growing (from \( 2^{x} \)) and decaying (from \( -2^{-x} \)) components.

Step 2: Determine Key Points

To graph the function, identify key points by evaluating \( f(x) \) at selected \( x \) values. \( f(0) = 2^{0} - 2^{0} = 0 \). Evaluate also at \( x = 1 \), \( f(1) = 2^{1} - 2^{-1} = 2 - \frac{1}{2} = \frac{3}{2} \). Finally, at \( x = -1 \), \( f(-1) = 2^{-1} - 2^{1} = \frac{1}{2} - 2 = -\frac{3}{2} \). These points are (0, 0), (1, \frac{3}{2}), and (-1, -\frac{3}{2}).

Step 3: Analyze Asymptotic Behavior

Exponential functions can have asymptotes. Consider \( 2^{-x} \) approaches 0 as \( x \) increases, meaning \( f(x) \) tends toward \( 2^{x} \). Similarly, \( 2^{x} \) approaches 0 as \( x \to -\infty \), making \( f(x) \to -2^{-x} \). These suggest an increasing function as \( x \to \infty \) and decreasing as \( x \to -\infty \).

Step 4: Plot Key Points and Sketch the Graph

On a graph, plot the calculated key points: (0, 0), (1, \frac{3}{2}), and (-1, -\frac{3}{2}). Use the asymptotic analysis to sketch a curve that increases toward \( +\infty \) to the right and decreases toward \( -\infty \) to the left, crossing the y-axis at the origin. The graph will appear as an S-shape, symmetric about the origin.

Key Concepts

Graphing Exponential Functions

When graphing exponential functions, you are dealing with a mathematical expression where the independent variable is the exponent of a constant base. This leads to rapid increases or decreases. In the given case of the function \( f(x) = 2^x - 2^{-x} \), we see an interplay of two exponentials:

• \( 2^x \) has exponential growth and rises sharply as \( x \) increases.
• \( 2^{-x} \) is equivalent to \( \frac{1}{2^x} \) and signifies exponential decay.

To effectively graph such functions, identify how these separate parts contribute to the overall graph. Ensure you have key points plotted and recognize areas of growth or decay. This highlights the dynamic nature of exponential functions.
Remember, the overall shape and direction depend highly on their components, which in this case result in an unexpected S-shaped form.

Asymptotic Behavior

Asymptotic behavior describes how a function behaves as it approaches a certain boundary, typically infinity. In our function \( f(x) = 2^x - 2^{-x} \), understanding when and how asymptotes occur will inform the graph’s structure:

• As \( x \to +\infty \), \( 2^{-x} \to 0 \) and the function approaches \( 2^x \), hinting at an unbounded increase.
• Conversely, as \( x \to -\infty \), \( 2^x \to 0 \), and the component \( -2^{-x} \) dominates, indicating an unbounded decrease.

Asymptotic analysis helps predict the behavior towards infinity or negative infinity, even if there are no actual asymptotes like horizontal or vertical lines. It provides insight into whether the function will ascend to infinity or descend, essential for sketching the limits of exponential functions.

Key Points in Graphing

Identifying key points is crucial for sketching a precise graph. For the function \( f(x) = 2^x - 2^{-x} \), determine these by plugging in strategic \( x \)-values:

• At \( x = 0 \): \( f(0) = 0 \), providing the origin (0, 0).
• At \( x = 1 \): \( f(1) = \frac{3}{2} \), a positive point (1, \( \frac{3}{2} \)).
• At \( x = -1 \): \( f(-1) = -\frac{3}{2} \), a negative point (-1, -\( \frac{3}{2} \)).

By marking these points, you create a framework that guides where the curve should bend or cross axes. These are essential anchors in the graph layout, helping convey how the function behaves and ensuring your sketch is accurate.

Exponential Growth and Decay

Exponential growth and decay are two sides of the same exponential process. In the function \( f(x) = 2^x - 2^{-x} \), each component exhibits these behaviors uniquely:

• \( 2^x \) indicates exponential growth, leading to rapid increase as \( x \) gets larger.
• \( 2^{-x} \), or \( \frac{1}{2^x} \), showcases exponential decay, quickly approaching zero as \( x \) increases.

These opposing forces within the expression govern the overarching behavior of our function. The growth component propels the curve upwards, while the decay pulls it down, creating the S-shaped graph you’d observe.
Understanding these principles of growth and decay is essential in predicting and adapting how various exponential functions will appear graphically across different values of \( x \).