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Understanding the domain and range of a function is crucial for correctly graphing it. The **domain** of a function includes all the possible input values (x-values) for which the function is defined. For the given exponential function, \( f(x) = -e^x \), the domain consists of all real numbers because exponentiation works for any real number x.
In interval notation, this is expressed as \((-∈fty, fty)\). So, you can plug any real number into the function, and it will give a corresponding y-value.

The **range**, on the other hand, refers to all possible output values (y-values). For \( f(x) = -e^x \), we start with the parent function \( e^x \), which only produces positive results. By adding a negative sign in front of \( e^x \), every positive output of \( e^x \) is converted to a negative output. Thus, the range of \( -e^x \) is all real numbers less than zero. In interval notation, the range is \(( -ft, 0) \). Essentially, the function will never produce zero or any positive number.

An **exponential function** is a mathematical function of the form \( f(x) = a \times b^x \) where \( a \) is a constant, \( b \) is the base greater than zero and not equal to one, and \( x \) is the exponent. In this exercise, our function is \( f(x) = -e^x \), where \( e \) is the base of natural logarithms, approximately equal to 2.71828. The base \( e \) is often used in natural growth processes.
Characteristics of an exponential function include:

• An **constant ratio of change** for equal increments in x-values.
• Rapid growth or decay; the function can grow large quickly or decrease towards zero depending on the base and the sign in front.

The parent function \( e^x \) has some standard properties, such as increasing rapidly and having the y-intercept at 1. Here, since we have \( -e^x \), you need to note the negative sign, which will reflect the graph across the x-axis, meaning it will decrease rapidly.

The term **function transformation** refers to modifying the basic parent function to reflect different shifts, stretches, shrinks, or reflections. For instance, let's consider the given function \( f(x) = -e^x \). One major transformation here is the **reflection**. The negative sign before \( e^x \) causes the graph to flip over the x-axis.

Here’s a breakdown of how transformations work:

• **Vertical Shifts**: Adding or subtracting a constant from \( f(x) \).
• **Horizontal Shifts**: Adding or subtracting a constant within the exponent in \( e^x \).
• **Vertical Stretching/Compressing**: Multiplying the function by a constant greater or less than 1.
• **Reflections**: Multiplying the function by -1.

In our function \( f(x) = -e^x \), the key transformation is the reflection. Notice how the original exponential growth behavior changes to exponential decay. Instead of the graph passing through (0,1) it now passes through (0,-1) and plunges downward as x increases. This is a visual representation of the inverse relationship caused by the reflective transformation.