Problem 38 - Precalculus with Calculus Previews

Question

Use \(f(-x)=f(x)\) to demonstrate that the given function is even. Sketch the graph of \(f\). $$ f(x)=e^{-|x|} $$

Step-by-Step Solution

Answer

The function \(f(x)=e^{-|x|}\) is even, as \(f(-x)=f(x)\). The graph is symmetric about the y-axis, starting at \(1\) at \(x=0\) with exponential decay on both sides.

Step 1: Understand Even Functions

A function is considered even if, for every value of \(x\) in the domain, \(f(-x) = f(x)\). This means the graph is symmetric about the y-axis.

Step 2: Analyze Given Function

The function given is \(f(x) = e^{-|x|}\). Note that the expression \(|x|\) represents the absolute value, which is always non-negative, making \(-|x|\) also negative or zero.

Step 3: Substitute \(-x\) into the Function

Substitute \(-x\) in place of \(x\) to get \(f(-x) = e^{-|-x|}\). It is important to note that \(|-x| = |x|\) since absolute value disregards sign.

Step 4: Simplify the Expression

Since \(|-x| = |x|\), \(f(-x) = e^{-|-x|} = e^{-|x|}\). So, \(f(-x) = f(x)\), which confirms that the function is even.

Step 5: Sketch the Graph of the Function

To sketch \(f(x) = e^{-|x|}\), consider the points: \((0, e^0 = 1)\), \((1, e^{-1})\), and \((-1, e^{-1})\). The graph is symmetric about the y-axis with an exponential decay from \(x = 0\) in both directions, illustrating even symmetry.

Key Concepts

Absolute Value in Functions

The absolute value of a number, denoted as \(|x|\), represents the number's distance from zero on the real number line. It is always non-negative, meaning the absolute value never results in a negative number. For example, \(|5| = 5\) and \(|-5| = 5\).
The concept of absolute value is crucial in understanding the function \(f(x) = e^{-|x|}\). In this function, \(-|x|\) translates the input to always be zero or negative. As a result, this function considers how far \(|x|\) is from zero, nullifying any negative signs.
When you substitute \(-x\) into \(f(x) = e^{-|x|}\), since \(|-x| = |x|\), you can simplify this expression to confirm symmetries, such as even functions, by removing the effect of any negative inputs.

Symmetric Graphs and Even Functions

Symmetric graphs display a mirror-like quality, which in mathematical terms, is often symmetry about the y-axis. \(f(x) = e^{-|x|}\) is an example of an even function because it holds the property that \(f(-x) = f(x)\) for all values in its domain.
Why are symmetric graphs significant? They help in understanding the behavior of functions in relation to their geometric representation. A symmetric graph about the y-axis will look the same on either side. By checking the symmetry, especially in even functions, you confirm that the same output results from both negative and positive input values.

• A graph symmetric about the y-axis means that any point \( (a, b) \) on the graph will have a mirrored point \((-a, b)\).
• This type of symmetry leads to easier analysis and prediction of function values based on its graphical properties.

Symmetry in even functions like \(f(x) = e^{-|x|}\) simplifies both calculation and visualization processes.

Exponential Decay Characteristics

Exponential decay is a process where quantities reduce at a rate proportional to their current value. In the function \(f(x) = e^{-|x|}\), the negative exponent indicates a decay property. This means as \(|x|\) increases, \(f(x)\) decreases.
Visualizing exponential decay involves imagining a downward sloping curve that approaches zero but never reaches it, characterizing by a rapid decrease that gradually slows. Here's how it plays out in our function:

• At \(x = 0\), the value is \(e^0 = 1\), the highest point on the graph.
• As \(|x|\) increases, beyond zero in either direction, the value \(e^{-|x|}\) shrinks, illustrating the decay process.
• The curve remains positive, consistently approaching zero, capturing the essence of decay without negatives.

Thus, the function \(f(x) = e^{-|x|}\) is not only even but also efficiently showcases the principles of exponential decay, ensuring the decay is symmetric and predictable.