91Ó°ÊÓ

When looking at exponential functions like \( y = -e^{-x} \), understanding the domain and range is essential. Here, the domain, or the set of all possible input values \( x \), includes all real numbers. This is a characteristic of all exponential functions as they are defined for any \( x \) value without any restrictions.
The range, on the other hand, refers to the set of all possible output values \( y \). In this particular function, since it is a negative exponential function, the range of \( y \) is less than zero. In simpler terms, \( y \) will never be positive and will always have a value less than or equal to zero. This is because \( e^{-x} \) is inherently positive and \( -e^{-x} \) simply inverts this.You can think of the range as the ceiling that limits how high \( y \) can go, which in this case caps at zero but doesn't reach it.

Whenever we graph a function, identifying its intercepts is a crucial step. Let's focus on the y-intercept first for the exponential function \( y = -e^{-x} \). The y-intercept occurs where the graph crosses the y-axis, which is when \( x = 0 \).By substituting \( x = 0 \) into the function, we get:

• \( y = -e^0 = -1 \)

This informs us that the only intercept where the graph touches any axis is at the point \( (0, -1) \). This signifies that as soon as \( x = 0 \), the value of \( y \) is exactly -1.
In terms of the x-intercept, the function \( y = -e^{-x} \) will never actually cross the x-axis. This is because, regardless of the value of \( x \), \( y \) never equals zero. Unlike some polynomial functions, exponential functions like this one result in a consistent trend without ever touching or crossing the x-axis.

An asymptote is essentially an invisible line that a graph approaches but never actually meets or crosses. In the function \( y = -e^{-x} \), you will observe a horizontal asymptote. This happens because as \( x \) tends to positive infinity, \( y \) draws nearer and nearer to zero without actually becoming zero. Thus, the horizontal asymptote here is the line \( y = 0 \).
Think of horizontal asymptotes as a sort of boundary that the graph comes infinitely close to, but never touches. For \( y = -e^{-x} \), the graph remains beneath this asymptote as all values generated by \( y \) are negative. This behavior accentuates the unique nature of exponential functions: no matter how far you extend along the x-axis, the exponential decay halts just shy of the asymptote but never quite reaches it.