A) \( f(x)=3^{x} \)

The function \( f(x) = 3^x \) is an exponential function where the base is 3. Here are some key properties and characteristics of this function: 1. **Domain**: The domain of \( f(x) \) is all real numbers, \( (-\infty, \infty) \). 2. **Range**: The range of \( f(x) \) is all positive real numbers, \( (0, \infty) \). 3. **Intercepts**: - **Y-intercept**: When \( x = 0 \), \( f(0) = 3^0 = 1 \). So, the y-intercept is at the point \( (0, 1) \). - **X-intercept**: There is no x-intercept since \( 3^x \) is never zero for any real number \( x \). 4. **Behavior**: - As \( x \to -\infty \), \( f(x) \to 0 \). - As \( x \to \infty \), \( f(x) \to \infty \). 5. **Increasing/Decreasing**: The function is strictly increasing for all \( x \) because the derivative \( f'(x) = 3^x \ln(3) \) is always positive. 6. **Asymptote**: The function has a horizontal asymptote at \( y = 0 \) as \( x \) approaches negative infinity. 7. **Graph**: The graph of \( f(x) = 3^x \) is a smooth curve that rises steeply as \( x \) increases and approaches the x-axis (but never touches it) as \( x \) decreases. If you have any specific questions or need further analysis regarding this function, feel free to ask!