1 The absolute value function is piecewise-defined because its behavior changes based on the sign of the input.
2 Exponential and logarithmic functions are inverses of each other.
3 Inverse trigonometric functions (arcsin, arccos, arctan) become functions when their domains are restricted.
4 In order to determine which model fits data better, examine the growth pattern: exponential for rapid growth, power for slower growth.
1 The defining characteristic is that different rules apply to different intervals of the domain, reflecting the need for different expressions depending on whether the input is negative, zero, or positive.
2 Exponential and logarithmic functions are inverses of each other. In essence, they "undo" each other's operations. Exponential functions involve repeated multiplication, while logarithmic functions reverse this process by finding the exponent needed to obtain a certain result.
3 For example, arcsin has a restricted domain of [-π/2, π/2], arccos has a domain of [0, π], and arctan has a domain of (-π/2, π/2). These restrictions ensure that the inverse functions pass the horizontal line test and are, therefore, functions.
4 Use logarithmic transformation or plotting to identify trends and choose the most suitable model.