1. **State the problem:** We need to find the domain and range of the function $$f(x) = 5e^{2x}$$. 2. **Recall the domain rules:** The domain of an exponential function $$e^{kx}$$ is all real numbers $$(-\infty, \infty)$$ because the exponential function is defined for every real input. 3. **Determine the domain:** Since $$f(x) = 5e^{2x}$$ is an exponential function multiplied by 5, the domain remains all real numbers: $$\text{Domain} = (-\infty, \infty)$$ 4. **Recall the range rules:** The exponential function $$e^{2x}$$ is always positive for all real $$x$$, i.e., $$e^{2x} > 0$$. 5. **Determine the range:** Multiplying by 5 (a positive constant) keeps the function positive. Therefore, $$f(x) = 5e^{2x} > 0$$ for all $$x$$. Hence, the range is: $$\text{Range} = (0, \infty)$$ **Final answer:** - Domain: $$(-\infty, \infty)$$ - Range: $$(0, \infty)$$