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To tackle the function \( f(x) = (x + 2)e^{-2} \), we can identify some interesting properties! First, let’s determine intercepts. The y-intercept occurs when \( x = 0 \) leading to \( f(0) = 2e^{-2} \). For the x-intercept, we set \( f(x) = 0 \), giving us \( x + 2 = 0 \), hence \( x = -2 \). Now, let’s look at its behavior: the function is linear since it can be expressed as \( f(x) = e^{-2} x + 2e^{-2} \). Being a linear function, it will have constant positive or negative behavior depending on the value of the slope \( e^{-2} \), which is always positive. Thus, \( f(x) \) is positive for \( x > -2 \) and negative for \( x < -2 \). On the limit fronts, as \( x \to \infty \), \( f(x) \to \infty \) and as \( x \to -\infty \), \( f(x) \to -\infty \). For asymptotes, there are none since it's a linear function; horizontal and vertical asymptotes apply to different function types. The first derivative \( f'(x) = e^{-2} \), indicating it doesn’t change—it’s positive everywhere. Therefore, the function is always increasing, confirming it would rise towards infinity. For further exploration, consider derivatives in connection to rates of change and motion—concepts beautifully illustrated in Graph Theory or Calculus. Delving into those areas can help illuminate how functions behave in real-world contexts, such as predicting population growth or analyzing financial trends.