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Graph f(x)=e^-|x|

Name: Problem 80, Chapter 3: Precalculus
Uploaded: 2020-08-10T17:50:16-08:00
Duration: 15 s
Channel: Katelyn Chen
Description: Graph f(x)=e^-|x|

Key Concepts

Piecewise Definition

A piecewise function indicates that the function can be expressed in separate rules depending on the input values. In the case of f(x)= e^(|x|), it can be seen as having one rule when x is non-negative (f(x)= e^x) and another when x is negative (f(x)= e^(-x)). This concept helps in understanding and graphing the function by considering each interval separately.

Even Function and Symmetry

A function is even if it satisfies the condition f(x) = f(-x) for all x in its domain, which implies symmetry about the y-axis. The use of an absolute value in the function's argument ensures that the graph is mirrored on either side of the y-axis, simplifying analysis and sketching of the function.

Absolute Value Transformation

The absolute value operation transforms any input into its non-negative counterpart, which directly affects the graph of a function. When applied inside another function, it modifies the input such that the output remains the same for both positive and negative values of x. This operation is essential for creating even functions and results in symmetry with respect to the y-axis.

Exponential Function

An exponential function, such as e^x, describes rapid growth or decay. It is characterized by a constant base raised to a variable exponent, and its graph typically passes through a key point like (0, 1) while approaching an asymptote (commonly the x-axis). This concept is fundamental in understanding many natural and technological processes that involve rate changes.

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