Gross Phillips
11/14/2023 · High School
f) \( f(x)=3^{x} \)
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Step-by-step Solution
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. **Monotonicity**: The function is strictly increasing for all \( x \) because the derivative \( f'(x) = 3^x \ln(3) \) is always positive.
6. **Asymptote**: The line \( y = 0 \) (the x-axis) is a horizontal asymptote.
7. **Graph**: The graph of \( f(x) = 3^x \) is a smooth curve that rises steeply as \( x \) increases and approaches the x-axis as \( x \) decreases.
If you have any specific questions or need further analysis regarding this function, feel free to ask!
Quick Answer
\( f(x) = 3^x \) is an exponential function with domain \( (-\infty, \infty) \), range \( (0, \infty) \), y-intercept at \( (0, 1) \), no x-intercept, and is strictly increasing.
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