4. sketch the graph of $f(x) = e^{-x}$.

Explanation:

Step1: Identify the function type

The function $f(x) = e^{-x}$ is a decreasing exponential function.

Step2: Determine the y-intercept

Evaluate the function at $x = 0$. $$f(0) = e^{0} = 1$$

Step3: Analyze horizontal asymptote

Determine the limit as $x$ approaches infinity. $$\lim_{x \to \infty} e^{-x} = 0$$

Step4: Analyze behavior as x decreases

Determine the limit as $x$ approaches negative infinity. $$\lim_{x \to -\infty} e^{-x} = \infty$$

Step5: Identify key points

Calculate values for $x = -1$ and $x = 1$. $$f(-1) = e^1 \approx 2.718, f(1) = e^{-1} \approx 0.368$$

Answer:

The graph of $f(x) = e^{-x}$ is a curve that passes through the y-intercept $(0, 1)$. It approaches the x-axis ($y = 0$) as a horizontal asymptote as $x \to \infty$ and increases rapidly toward $\infty$ as $x \to -\infty$. The curve is always positive and strictly decreasing.