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The absolute value function is a mathematical tool that transforms negative numbers into positive numbers, while leaving positive numbers unchanged. It is often represented by vertical bars: \( |x| \).
The absolute value of a number \( x \) is its distance from zero on the number line. For example, the absolute value of both -3 and 3 is the same: 3.

• For any positive number \( x \), \( |x| = x \).
• For any negative number \( x \), \( |x| = -x \).
• For zero, \( |0| = 0 \).

The role of the absolute value in the function \( f(x) = e^{|x|} \) influences how you interpret the function. When we calculate \( f(-x) \), the absolute value ensures that \( |-x| = |x| \), making the function independent of whether \( x \) is negative or positive.

An exponential function is a type of mathematical function where a constant base is raised to a variable exponent. The general form is \( f(x) = a^{x} \) where \( a \) is a positive constant. It's a crucial function in mathematics due to its rapid growth properties.
In this context, we consider \( f(x) = e^{|x|} \), which combines exponential function with absolute value. The base \( e \) is a unique constant approximately equal to 2.71828, known as Euler's number, and is the foundation of natural logarithms.

• Exponential functions grow very fast as \( x \) increases.
• The function \( e^{|x|} \) will produce identical results for \( x \) and \(-x \).
• As \( |x| \) increases, \( e^{|x|} \) tends toward infinity.

This function is notable because its rate of growth is proportional to its current value, making it suitable for modeling phenomena with constant relative growth rates, such as interest compounding and population growth.

The domain of a function refers to all possible input values (\( x \)) that the function can accept without resulting in any undefined or forbidden operations. Generally, you define it for the kinds of numbers that you can use in real-world applications.
For the function \( f(x) = e^{|x|} \), the domain includes all real numbers because:

• Absolute value \( |x| \) is defined for any real number \( x \).
• The exponential function \( e^{x} \) is defined for all real numbers as well.

This comprehensive domain means that you can plug in any real number, positive, negative, or zero, into \( f(x) = e^{|x|} \) without encountering any mathematical difficulties.
Understanding the domain is essential for evaluating the range and potential behavior of the function, making it a foundational element in analyzing its properties.