UNIT 6%0D%0AFunctions%0D%0ALESSON 8%0D%0AProperties of Functions
In this lesson, we will explore various properties of functions. Functions are mathematical objects that describe the relationship between inputs and outputs. Understanding their properties can help us analyze and manipulate them more effectively.
1. Domain: The domain of a function is the set of all possible input values (or independent variables) for which the function is defined. It represents the "allowable" inputs for the function.
2. Range: The range of a function is the set of all possible output values (or dependent variables) that the function can produce. It represents the "possible" outputs for the function.
3. Mapping: A function can be represented as a mapping between input values and output values. This mapping can be shown using a graph, a table, or an equation.
4. One-to-one correspondence: A function is said to have a one-to-one correspondence if each input value corresponds to a unique output value, and each output value corresponds to a unique input value. This property ensures that there are no repetitions in the outputs for different inputs.
5. Continuous vs. Discontinuous: A function is continuous if its graph can be drawn without any breaks, jumps, or holes. On the other hand, a function is discontinuous if there are breaks, jumps, or holes in its graph.
6. Even and odd functions: A function is said to be even if it is symmetric with respect to the y-axis, meaning that f(x) = f(-x) for all x in the domain. In contrast, a function is odd if it is symmetric with respect to the origin, meaning that f(x) = -f(-x) for all x in the domain.
7. Monotonicity: A function is said to be increasing if its output values increase as the input values increase. Conversely, a function is said to be decreasing if its output values decrease as the input values increase.
8. Periodicity: A function is periodic if it repeats its values over regular intervals. The smallest positive value of x for which f(x) = f(x + p), where p is the period, is called the fundamental period of the function.
These are some of the key properties of functions that can help us understand and analyze their behavior. By studying these properties, we can make predictions about the behavior of functions and use them in various applications.