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Exponential functions are a type of mathematical expression where the variable is the exponent of a constant base. Typically, they take the form \(y = a \, b^x\), where \(b\) is a positive constant and \(a\) is a coefficient. These functions showcase rapid change, which is why they appear frequently in scenarios involving growth or decay.

In the case of \(y = -e^x\), this is an exponential function but with a slight twist - the base is the natural constant \(e\), and the coefficient is negative. This negative coefficient flips the usual upward growth behavior of \(e^x\) into a decline.

• The function continuously decreases as \(x\) moves from left to right.
• Instead of rising into infinity, the graph heads toward zero, never reaching it because of its nature.

This behavior is crucial for understanding how the function interacts with its environment - a set of properties that define its overall behavior.

The domain of an exponential function describes all possible input values (\(x\) values) for which the function is defined. For the function \(y = -e^x\), the domain is all real numbers because you can plug any real \(x\) value into an exponential function without restrictions. Thus, it covers the continuous interval from \(-\infty\) to \(\infty\).

• Domain: \((-\infty, \infty)\)

Regarding range, it represents all possible output values (\(y\) values). For \(y = -e^x\), this is noteworthy. Normally, \(e^x\) is positive for all \(x\), making \(-e^x\) a negative value of \(e^x\). Therefore, \(y\) never becomes positive, defining the range as all numbers below zero.

• Range: \((-\infty, 0)\)

This step of identifying domain and range helps in clearly visualizing the scope of the function, ensuring you're considering all possible outputs and scenarios in its graph.

Intercepts refer to the points where the graph of the function crosses the axes. For the function \(y = -e^x\), major intercepts to consider are:

• Y-Intercept: To find it, set \(x = 0\). Substituting in, \(y = -e^0 = -1\). Thus, the y-intercept is at \((0, -1)\).
• X-Intercept: An x-intercept would occur where \(y = 0\). However, \( -e^x\) is never zero for any real \(x\), meaning no x-intercepts exist.

Asymptotes are lines that the graph approaches but never actually reaches. In the case of \(y = -e^x\), there's a horizontal asymptote due to its limit of approaching zero. Here, as \(x\) moves towards infinity, \(-e^x\) gets closer and closer to zero without ever touching or crossing it.

• The horizontal asymptote is the line \(y = 0\).

Recognizing intercepts and asymptotes provides insight into the behavior and boundaries of the curve, essential for accurately sketching or analyzing the graph of an exponential function.