E to The Negative X Calculator

What is e to the negative x?

The expression e^-x represents the exponential function with base e (approximately 2.71828) raised to the power of -x. This function is the mathematical inverse of e^x and has several important properties:

• It's always positive for all real x
• It decreases rapidly as x increases
• It approaches zero as x approaches infinity
• It's the solution to certain differential equations

This function is particularly important in probability theory, where it appears in the exponential distribution, and in physics, where it describes radioactive decay and other exponential decay processes.

Formula

This function is continuous and differentiable everywhere, making it useful in calculus and differential equations.

How to use this calculator

1. Enter the value of x in the input field
2. Click the "Calculate" button
3. View the result in the result panel
4. Optionally view the chart showing the function behavior

Interpreting results

The value of e^-x represents:

• For x = 0: e⁰ = 1
• For x > 0: The value decreases towards 0
• For x < 0: The value increases exponentially

This function is particularly useful in modeling processes where quantities decrease exponentially over time, such as radioactive decay or heat dissipation.

Examples

Example 1: Positive exponent

If x = 1:

e^-1 ≈ 0.3679

This represents a 63.2% decrease from the initial value.

Example 2: Zero exponent

If x = 0:

e⁰ = 1

This is the base case for exponential functions.

Example 3: Negative exponent

If x = -1:

e¹ ≈ 2.7183

This shows exponential growth.