Problem 76 - Precalculus

Question

$$\text { Graph } f(x)=e^{-|x|}$$

Step-by-Step Solution

Answer

The graph of \( f(x) = e^{-|x|} \) is symmetric about the \( y \)-axis, decreasing from (0,1) towards the \( x \)-axis on both sides rapidly.

Step 1: Understand the Function

The function given is \( f(x) = e^{-|x|} \). It involves the natural exponential function and the absolute value of \( x \). The expression \( -|x| \) ensures the exponent is always non-positive since \( |x| \) is always non-negative.

Step 2: Analyze the Symmetry

The function \( f(x) = e^{-|x|} \) is symmetric around the \( y \)-axis. This is because \( |x| = |-x| \), leading to \( f(x) = f(-x) \). This symmetry indicates that the graph will look the same on the left and right sides of the \( y \)-axis.

Step 3: Evaluate at Key Points

Evaluate the function at a few key points. At \( x = 0 \), \( f(0) = e^{0} = 1 \). For \( x = 1 \), \( f(1) = e^{-1} \approx 0.3679 \). For \( x = 2 \), \( f(2) = e^{-2} \approx 0.1353 \). This pattern continues, diminishing as \( x \) moves farther from zero.

Step 4: Sketch the Graph

Based on the assessments:- The graph passes through the point (0,1).- As \( x \) becomes positive or negative, \( f(x) \) approaches zero (i.e., the \( x \)-axis), never touching it.- The graph decreases rapidly due to the exponential decay factor \( e^{-|x|} \).Plotting these points and their mirrored counterparts across the \( y \)-axis, sketch a curve that starts at (0,1) and races towards the x-axis on either side.

Key Concepts

Absolute Value

The concept of absolute value plays a crucial role when graphing exponential functions like \( f(x) = e^{-|x|} \). The absolute value of a number is its distance from zero on the number line, regardless of direction. This means that \( |x| \) is always non-negative, which directly affects our given function since the exponent becomes non-positive.

In simpler terms, for any real number \( x \), \(|x| \) is equal to \( x \) if \( x \) is positive or zero, and \(-x\) if \( x \) is negative. The equation \( f(x) = e^{-|x|} \) thus ensures that we're working with non-positive exponents, which is essential when understanding the shape of the graph.

• Positive \( x \) values: \( -|x| \) becomes \( -x \), so the function becomes \( e^{-x} \).
• Negative \( x \) values: \( -|x| \) becomes \( x \), again leading to \( e^{-x} \) due to the absolute value.

Exponential Decay

Exponential decay describes a process where a quantity decreases at a rate proportional to its current value. In the function \( f(x) = e^{-|x|} \), exponential decay is depicted as \( x \) moves away from zero. The primary reason for this behavior is the negative exponent, \(-|x|\), which ensures that the values of \( f(x) \) decrease quickly towards zero, but never truly reach it.

What this means for the graph is:

• The function value drops rapidly as \( x \) deviates from zero, illustrating the concept of decay.
• At \( x = 0 \), the function peaks at \( f(0) = 1 \).
• For positive and negative values of \( x \) alike, the curve approaches the x-axis, but remains asymptotic, synonymous with exponential decay.

Understanding this concept allows you to predict the behavior of various real-world scenarios, like cooling temperatures or radioactive decay, where the rate of decrease is always relative to the present amount.

Symmetry in Graphs

Symmetry in graphs simplifies how we interpret and sketch functions. For the function \( f(x) = e^{-|x|} \), symmetry is evident through its invariance to reflection over the \( y \)-axis. This means that every point on one side of the \( y \)-axis has a corresponding mirror point on the opposite side. As a consequence:

• The graph effectively reflects one side onto the other, simplifying the plotting process.
• The property \( f(x) = f(-x) \) showcases this mirroring effect, meaning the graph creates a balanced, elegant curve when plotted.

This symmetry gives rise to a predictable pattern over the \( y \)-axis. It means you only need to calculate or draw one half of the graph to understand its full shape.

In practical terms, symmetry in functions like these allows for easier calculations, as changes on one side of the graph will be mirrored exactly, reducing complexity in visualizing how the function behaves.