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Magazine Chapter 10: Problem 45
Graph each of the exponential functions. $$ f(x)=2^{x}-2^{-x} $$
Short Answer
Expert verified
Graph is an S-shaped curve passing through (0, 0), (1, 1.5), (-1, -1.5).
Step by step solution
01
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.
02
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}).
03
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 \( x \to -\infty \).
04
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
Remember, the overall shape and direction depend highly on their components, which in this case result in an unexpected S-shaped form.
- \( 2^x \) has exponential growth and rises sharply as \( x \) increases.
- \( 2^{-x} \) is equivalent to \( \frac{1}{2^x} \) and signifies exponential decay.
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鈥檚 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.
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, -\( \frac{3}{2} \)).
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:
Understanding these principles of growth and decay is essential in predicting and adapting how various exponential functions will appear graphically across different values of \( x \).
- \( 2^x \) 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.
Understanding these principles of growth and decay is essential in predicting and adapting how various exponential functions will appear graphically across different values of \( x \).
