Problem 26 - Intermediate Algebra

Question

Graph each function. $$ f(x)=-e^{x} $$

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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.

Step 3: 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.

Step 4: 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.

Step 5: 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 \( f(x) = -e^x \).

Key Concepts

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 \( f(-1) = -0.368 \).

The plotted points give a sense of the curve's direction and behavior. Connect these points smoothly, making sure the graph approaches the horizontal asymptote as \( x \) moves towards positive infinity. The key is to maintain precision in reflecting the base form to capture the inverted curve.

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.

The key takeaway is the function's exponential decay in the negative direction, highlighting the inverse growth pattern due to the negative sign.

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.

Understanding \( e \) not only helps in graphing but also in making sense of the mathematics behind real-life growth patterns and applications. This constant plays a pivotal role in numerous calculations, serving as a cornerstone of calculus and higher mathematical analysis.