91Ó°ÊÓ

The domain and range of a function describe the possible values for the input (domain) and the output (range). For our function, \(f(x) = -e^{-x}\), let's break down these two concepts.

Domain: The domain of a function is the set of all possible values for the variable \(x\). In the case of \(f(x) = -e^{-x}\), there are no restrictions on the value of \(x\). Hence, the domain is all real numbers: \( (-\text{\infty}, \text{\infty}) \).

Range: The range is the set of all possible values of \(f(x)\). As \(x\) gets very large (\(x \to \text{\infty}\)), \(e^{-x}\) approaches 0, making \(f(x)\) approach \(0\) from below. On the other hand, as \(x\) becomes very negative (\(x \to - \text{\infty}\)), \(e^{-x}\) becomes very large, causing \(f(x)\) to approach \(- \text{\infty}\). Hence, the range is \( (-\text{\infty}, 0) \).

Exponential functions are defined by equations like \(y = e^{x}\) or \(y = e^{-x}\). These functions are named due to their characteristic shape, which involves rapid growth or decay.

The function \(f(x) = -e^{-x}\) is an exponential decay function but with a twist: the negative sign in front of \(e^{-x}\).

• As \(x\) increases, \(e^{-x}\) quickly approaches \(0\). The function \(f(x) = -e^{-x}\) therefore approaches \(0\) as \(x\) becomes very large.
• As \(x\) decreases, \(e^{-x}\) grows very quickly. Multiplying this growth by \(-1\) causes \(f(x)\) to approach \(-\text{\infty}\).

Understanding the basic behavior of exponential functions helps in analyzing their graphs. For \(f(x) = -e^{-x}\):

• \(f(x)\) decreases to approach \(0\) as \(x\) increases
• \(f(x)\) plummets to \(- \text{\infty}\) as \(x\) decreases.

The 'decay rate' in exponential functions describes how quickly the function's values decline. For \(f(x) = -e^{-x}\), the decay rate is driven by the \(-e^{-x}\) term.

In general, for an exponential function \(g(x) = e^{-kx}\), \(k\) represents the rate at which the function decays. In \(f(x)\), the rate \(k\) is \(1\), indicating a standard exponential decay.

Key points to remember about the decay rate in \(f(x) = -e^{-x}\):

• The most significant change in \(f(x)\) happens when \(x\) is close to \(0\). This is where the curve is steepest.
• As \(x\) moves away from \(0\) towards positive infinity, the change in \(f(x)\) slows down.
• As \(x\) moves towards negative infinity, \(f(x)\) rapidly decreases towards \(-\text{\infty}\), showing the decay rate is constant.