Question
short short define exponential,exponential decay,domain,range
Answers
GPT-4o mini
- **Exponential**: A mathematical function where a constant base is raised to a variable exponent, generally in the form \( f(x) = a \cdot b^x \), where \( a \) is a constant, \( b \) is the base (a positive real number), and \( x \) is the exponent.
- **Exponential Decay**: A process where a quantity decreases at a rate proportional to its current value, often described by the function \( f(t) = a \cdot e^{-kt} \), with \( a > 0 \) and \( k > 0 \), resulting in rapid initial decreases that slow over time.
- **Domain**: The set of input values (or x-values) for which a function is defined. For exponential functions, the domain is typically all real numbers (\( (-\infty, \infty) \)).
- **Range**: The set of output values (or y-values) that a function can take. For exponential functions, the range is usually \( (0, \infty) \) for the \( f(x) = a \cdot b^x \) form, with \( a > 0 \).
- **Exponential Decay**: A process where a quantity decreases at a rate proportional to its current value, often described by the function \( f(t) = a \cdot e^{-kt} \), with \( a > 0 \) and \( k > 0 \), resulting in rapid initial decreases that slow over time.
- **Domain**: The set of input values (or x-values) for which a function is defined. For exponential functions, the domain is typically all real numbers (\( (-\infty, \infty) \)).
- **Range**: The set of output values (or y-values) that a function can take. For exponential functions, the range is usually \( (0, \infty) \) for the \( f(x) = a \cdot b^x \) form, with \( a > 0 \).