1. **Problem:** Graph the function $f(x) = e^x$ and find its domain and range. 2. **Formula and rules:** The function is an exponential function of the form $f(x) = e^x$, where $e$ is Euler's number approximately equal to 2.718. 3. **Domain:** The domain of $f(x) = e^x$ is all real numbers because you can plug any real number into the exponent. 4. **Range:** Since $e^x$ is always positive and never zero, the range is $(0, \infty)$. 5. **Graph shape:** The graph is an increasing curve starting near zero for large negative $x$ and growing rapidly for positive $x$. --- 1. **Problem:** Graph the function $f(x) = e^{x-2}$ and find its domain and range. 2. **Formula and rules:** This is a horizontal shift of the standard exponential function by 2 units to the right. 3. **Domain:** All real numbers. 4. **Range:** $(0, \infty)$, same as the standard exponential. 5. **Graph shape:** Same shape as $e^x$ but shifted right by 2. --- 1. **Problem:** Graph the function $f(x) = e^x - 2$ and find its domain and range. 2. **Formula and rules:** This is a vertical shift down by 2 units of the standard exponential. 3. **Domain:** All real numbers. 4. **Range:** Since $e^x > 0$, $e^x - 2 > -2$, so range is $(-2, \infty)$. 5. **Graph shape:** Same shape as $e^x$ but shifted down by 2. --- 1. **Problem:** Graph the function $f(x) = e^{-x}$ and find its domain and range. 2. **Formula and rules:** This is an exponential decay function because of the negative exponent. 3. **Domain:** All real numbers. 4. **Range:** $(0, \infty)$. 5. **Graph shape:** Decreasing curve starting high at $x \to -\infty$ and approaching zero as $x \to \infty$. --- 1. **Problem:** Graph the function $f(x) = e^x - 10$ and find its domain and range. 2. **Formula and rules:** Vertical shift down by 10 units of the standard exponential. 3. **Domain:** All real numbers. 4. **Range:** Since $e^x > 0$, $e^x - 10 > -10$, so range is $(-10, \infty)$. 5. **Graph shape:** Same shape as $e^x$ but shifted down by 10.