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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 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 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.