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Understanding the domain and range of a function is essential in graphing. The domain refers to all possible input values (or x-values) that a function can accept. For the exponential function \( y = e^{-x} - 1 \), its domain is all real numbers \((-\infty, \infty)\). This is because exponential functions are defined everywhere on the real number line:

• There are no restrictions on the x-values for \( e^{-x} \).
• No operations like division by zero or square roots of negative numbers occur.

The range of a function is all possible output values (or y-values). Initially, for \( y = e^{-x} \), the output values are \((0, \infty)\), as the function approaches zero but never touches it. Shifting the function downward by 1 changes the range:

• The whole graph moves one unit down.
• This adjustment changes the range to \((-1, \infty)\).

This means the smallest value of \( y \) is now just slightly above \(-1\), and it can grow without bound.

Transformations change the appearance of functions without altering their basic nature. For the function \( y = e^{-x} - 1 \), two transformations are applied to the base exponential function \( y = e^x \):

• First, an exponential reflection occurs over the y-axis, transforming \( y = e^x \) into \( y = e^{-x} \). A reflection across the y-axis affects the x-values by making them negative, which turns the increasing graph of \( e^x \) into the decreasing graph of \( e^{-x} \).
• Second, the function is shifted down by 1 unit, modifying it to \( y = e^{-x} - 1 \). Vertical shifts adjust the y-values directly by adding or subtracting from them. Here, subtracting 1 lowers every point on the graph by one unit, impacting the range and aligning the asymptote at \( y = -1 \).

Such transformations allow us to model real-world situations by adjusting a standard function to fit specific conditions.

An asymptote is a line that a graph approaches but never actually reaches. In the function \( y = e^{-x} - 1 \), the horizontal asymptote is crucial for understanding the behavior of the graph as \( x \) becomes very large. As \( x \to \infty \), the expression \( e^{-x} \to 0 \), causing the entire function \( y = e^{-x} - 1 \) to approach \( -1 \).

• The line \( y = -1 \) acts like a barrier the function draws near to but never intersects.
• This horizontal asymptote illustrates that despite the diminishing value of \( e^{-x} \), \( y \) will always be slightly greater than \(-1\).

Understanding asymptotes helps in predicting the long-term behavior of functions and in sketching more accurate graphs, as these are the guidelines shaping the curve's path.