91Ó°ÊÓ

When dealing with the function \( f(x) = -e^x \), it's important to understand the domain and range concepts. The **domain** is the set of all possible input values, or \( x \)-values, that a function can accept. For our function \( f(x) = -e^x \), the base function \( y = e^x \) is defined for every real number. Therefore, the domain of \( f(x) = -e^x \) remains all real numbers, represented as \(( -\infty, \infty) \).
On the other hand, the **range** is the set of possible output values, or \( y \)-values, a function can produce. The range for the base function \( y = e^x \) includes all positive numbers, \(( 0, \infty) \). However, since \( f(x) = -e^x \) is a reflection over the x-axis, we multiply the \( y \)-values by \(-1\). This transformation flips the range to all negative numbers, which we write as \( (-\infty, 0) \).
In summary, for \( f(x) = -e^x \):

• Domain is \(( -\infty, \infty) \)
• Range is \( (-\infty, 0) \)

Function transformations involve altering the appearance of the graph of a function. There are various types of transformations like translations, reflections, stretching, and compressing. For our function \( f(x) = -e^x \), the transformation involved is a **vertical reflection**.
To understand it better, let's look at what a reflection means. A reflection in the x-axis changes the sign of the \( y \)-values. If you start with \( y = e^x \), where \( y \) is always positive because \( e^x > 0 \), multiplying by \(-1\) in \( f(x) = -e^x \) makes all \( y \)-values negative.
Thus, the reflection across the x-axis inherits the original shape of \( y = e^x \), but now all points are flipped to be below the x-axis. This has a significant visual impact, changing the curve's direction and orientation, but not its overall shape.
In terms of our problem:

• The vertical reflection results in the range changing from \((0, \infty)\) to \((-\infty, 0)\)
• The domain remains unaffected by this reflection.

Understanding horizontal asymptotes is crucial when graphing exponential functions. A **horizontal asymptote** is a line that the graph of a function approaches but never touches or crosses directly. It represents a boundary for the values in a graph. In our base function \( y = e^x \), as \( x \) approaches negative infinity, the function approaches 0 but never actually reaches it, creating a horizontal asymptote along the line \( y = 0 \).
The given function, \( f(x) = -e^x \), also approaches the same line as \( x \) moves towards negative infinity. Despite the vertical reflection that changes the function values to negative, the asymptote remains unaffected because reflecting across the x-axis does not adjust the \( x \)-values. The vertical reflection translates to \( f(x) = -e^x \) approaching the asymptote \( y = 0 \) from below.
Key points to remember about horizontal asymptotes for \( f(x) = -e^x \):

• The horizontal asymptote is \( y = 0 \).
• The curve approaches this line from below the x-axis as \( x \to -\infty \).
• This asymptotic behavior emphasizes that in transformations, the \( x \) values determine the asymptote, which would remain unchanged by vertical transformations like reflections.