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Exponential decay occurs when a quantity decreases rapidly at first and then levels off over time. In the given function, \( f(x) = -e^{-x} \), notice the exponent has a negative sign. This negative exponent signifies that as \( x \) increases, the value of \( e^{-x} \) gets smaller.

The negative sign in front of the exponential function further changes its behavior. Generally, functions of the form \( e^{-x} \) decay towards zero. However, multiplying by -1 inverts its direction. Instead of approaching zero from above, \( -e^{-x} \) approaches zero from below, creating a downward curve.

For example:

• At \( x = 0 \), \( f(0) = -e^0 = -1 \)
• At \( x = 1 \), \( f(1) = -e^{-1} = -\frac{1}{e} \)
• At \( x = -1 \), \( f(-1) = -e \)

These points highlight that as \( x \) increases, the function decreases but never quite reaches zero.

The domain and range are essential to understanding the behavior of functions. The domain refers to all possible input values (\( x \)-values) a function can accept, while the range refers to all possible output values (\( y \)-values) a function can produce.

For \( f(x) = -e^{-x} \), let's determine each:

• Domain: The function \( e^x \) and hence \( e^{-x} \) is defined for all real numbers. Therefore, the domain of \( f(x) \) is all real numbers, written as \( (-\infty, \infty) \).
• Range: As \( x \) moves from \( -\infty \) to \( \infty \), the output of \( e^{-x} \) varies. Multiplying by -1 reverses and shifts these values. Thus, \( f(x) = -e^{-x} \) ranges from approaching 0 (but never reaching it) to negatively approaching \( -\infty \). Thus, the range is \( (-\infty, 0) \).

Grasping these concepts ensures you understand how values within a function are mapped.

Function graphing visually represents the relationship between inputs and outputs of a function. For \( f(x) = -e^{-x} \), graphing helps illustrate its behavior and features.

Start by plotting key points:

• When \( x = 0 \), \( f(0) = -1 \)
• When \( x = 1 \), \( f(1) = -\frac{1}{e} \)
• When \( x = -1 \), \( f(-1) = -e \)

Next, sketch the general shape:

• This curve starts at \( -\infty \) when \( x \) is very small, as \( e^{-(-\infty)} = e^{\infty} = \infty \) and \( -e^{-\infty} = -\infty \).
• At \( x = 0 \), it passes through \( -1 \).
• As \( x \) increases, the function approaches 0 from below, never actually touching the x-axis.

Understanding and interpreting these points will help graph and comprehend the given function's behavior.