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Magazine Chapter 9: Problem 26
Graph each function. $$ f(x)=-e^{x} $$
Short Answer
Expert verified
The graph of \( f(x) = -e^x \) is an inverted exponential curve descending towards negative infinity on the left and approaching 0 on the right.
Step by step solution
01
Understand the Function
The given function is \( f(x) = -e^x \). This indicates an exponential function where the base is Euler's number \( e \) (approximately 2.71828), and the entire expression is multiplied by -1. This multiplication inverts the standard shape of the exponential graph.
02
Identify Key Characteristics
The exponential function \( e^x \) normally passes through the point (0,1), but due to the negative sign, the graph of \( f(x) = -e^x \) will pass through (0,-1). The curve will approach 0 asymptotically as \( x \) approaches positive infinity, and will decrease rapidly to negative infinity as \( x \) goes to negative infinity.
03
Plot Key Points
Select a few key values of \( x \) to plot points that define the curve: \((x, f(x))=(0, -1), (1, -e), (-1, -\frac{1}{e}) \). These points can be calculated: \(-e \) is approximately -2.718 and \(-\frac{1}{e} \) is approximately -0.368.
04
Sketch the Graph
Draw the y-axis and x-axis. Plot the calculated points: (0,-1), (1,-2.718), and (-1, -0.368). Draw the curve passing through these points such that it approaches 0 as \( x \) goes to positive infinity, and descends steeply as \( x \) goes towards negative infinity.
05
Analyze the Graph
The graph is decreasing for all \( x \), as the negative exponent causes the exponential function to decay instead of grow. It has a horizontal asymptote at \( y = 0 \). Use this visualization to interpret the behavior of \( f(x) = -e^x \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graphing Functions
When it comes to graphing functions, especially exponential functions, starting with a clear understanding of their basic forms is key. For an exponential function like \( f(x) = -e^x \), the approach includes identifying its base pattern and then applying transformations. Here, the graph of the base function \( e^x \) is reflected over the x-axis due to the multiplication by -1.
- The function \( f(x) = e^x \) typically passes through the point (0,1), but \( f(x) = -e^x \) changes this to (0,-1).
- As we choose values of \( x \) to calculate specific points, we get critical values like \( f(0) = -1 \), \( f(1) = -2.718 \), and \( f(-1) = -0.368 \).
Asymptotic Behavior
Asymptotic behavior describes how a function behaves as it approaches a boundary infinitely. For \( f(x) = -e^x \), this is crucial to understanding the graph.
- Here, the curve approaches \( y = 0 \) as \( x \) tends to positive infinity. This means that as \( x \) gets larger, the value of \( f(x) \) gets closer and closer to 0, but never actually reaches it.
- This behavior suggests a horizontal asymptote at \( y = 0 \), acting as a boundary line that the curve hovers near but does not cross.
- As \( x \) takes negative values, the function's value decreases without bound, moving towards negative infinity. This indicates the graph descends steeply.
Euler's Number
Euler's number, denoted as \( e \), is a fundamental mathematical constant approximately equal to 2.71828. It is the base of the natural logarithm and is found in various natural growth processes.
- In mathematics, \( e \) signifies the idea of growth at a constant proportion, like continuously compounding interest.
- Exponential functions based on \( e \), such as \( e^x \), serve to model real-world phenomena that grow or decay at consistent rates.
- When \( f(x) = -e^x \) incorporates Euler's number, it means we're dealing with a reflection, making the function an example of exponential decay rather than growth.
