Question
What are key feature of a exponential parent function
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GPT-4o mini
The exponential parent function is typically expressed in the form \( f(x) = a \cdot b^x \), where:
- \( a \) is a constant that represents the vertical stretch or compression and the direction of the graph (positive or negative).
- \( b \) is the base of the exponential function, which must be a positive real number. If \( b > 1 \), the function represents exponential growth. If \( 0 < b < 1 \), it represents exponential decay.
- \( x \) is the variable or exponent.
Here are some key features of the exponential parent function:
1. **Domain**: The domain of the exponential function is all real numbers, \( (-\infty, \infty) \).
2. **Range**: The range is \( (0, \infty) \) if \( a > 0 \) (the function approaches 0 but never reaches it). If \( a < 0 \), the range is \( (-\infty, 0) \).
3. **Intercept**: The function has a y-intercept at \( (0, a) \). For the basic exponential function \( f(x) = b^x \), this point would be \( (0, 1) \) if \( a = 1 \).
4. **Asymptote**: The horizontal line \( y = 0 \) (the x-axis) is a horizontal asymptote of the function. The graph approaches the asymptote but never touches or crosses it.
5. **Behavior**:
- **For \( b > 1 \)**: The function increases rapidly as \( x \) increases, showing exponential growth. As \( x \) approaches negative infinity, \( f(x) \) approaches 0.
- **For \( 0 < b < 1 \)**: The function decreases rapidly as \( x \) increases, showing exponential decay. As \( x \) approaches negative infinity, \( f(x) \) approaches infinity.
6. **Continuous and Smooth**: The exponential function is continuous and does not have any breaks, holes, or sharp turns in its graph.
7. **Differentiability**: The function is differentiable for all real numbers, and its derivative is also an exponential function, which contributes to its unique growth/decay characteristics.
These features help characterize the behavior and applications of exponential functions in various fields such as biology, finance, and physics, where processes often grow or decay exponentially.
- \( a \) is a constant that represents the vertical stretch or compression and the direction of the graph (positive or negative).
- \( b \) is the base of the exponential function, which must be a positive real number. If \( b > 1 \), the function represents exponential growth. If \( 0 < b < 1 \), it represents exponential decay.
- \( x \) is the variable or exponent.
Here are some key features of the exponential parent function:
1. **Domain**: The domain of the exponential function is all real numbers, \( (-\infty, \infty) \).
2. **Range**: The range is \( (0, \infty) \) if \( a > 0 \) (the function approaches 0 but never reaches it). If \( a < 0 \), the range is \( (-\infty, 0) \).
3. **Intercept**: The function has a y-intercept at \( (0, a) \). For the basic exponential function \( f(x) = b^x \), this point would be \( (0, 1) \) if \( a = 1 \).
4. **Asymptote**: The horizontal line \( y = 0 \) (the x-axis) is a horizontal asymptote of the function. The graph approaches the asymptote but never touches or crosses it.
5. **Behavior**:
- **For \( b > 1 \)**: The function increases rapidly as \( x \) increases, showing exponential growth. As \( x \) approaches negative infinity, \( f(x) \) approaches 0.
- **For \( 0 < b < 1 \)**: The function decreases rapidly as \( x \) increases, showing exponential decay. As \( x \) approaches negative infinity, \( f(x) \) approaches infinity.
6. **Continuous and Smooth**: The exponential function is continuous and does not have any breaks, holes, or sharp turns in its graph.
7. **Differentiability**: The function is differentiable for all real numbers, and its derivative is also an exponential function, which contributes to its unique growth/decay characteristics.
These features help characterize the behavior and applications of exponential functions in various fields such as biology, finance, and physics, where processes often grow or decay exponentially.
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