LESSON 8 Properties of Functions
In this lesson, we will discuss the properties of functions, which are mathematical tools used to describe relationships between variables.
1. Domain: The domain of a function is the set of all possible input values for the function. It defines the values for which the function is defined and meaningful. For example, the function f(x) = 1/x has a domain of all real numbers except x = 0, since division by zero is undefined.
2. Range: The range of a function is the set of all possible output values for the function. It represents the set of values that the function can produce. For example, the function f(x) = x^2 has a range of all non-negative real numbers, since squaring any real number yields a non-negative value.
3. One-to-one: A function is said to be one-to-one if each input value corresponds to a unique output value. In other words, no two different inputs produce the same output. For example, the function f(x) = x + 1 is one-to-one since for any two distinct values of x, the output values will also be distinct.
4. Onto: A function is said to be onto if every possible output value is mapped to by at least one input value. In other words, the range of the function is equal to its codomain. For example, the function f(x) = x^2 is not onto since it can never produce negative output values.
5. 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 every value of x in the domain. For example, the function f(x) = x^2 is even since f(x) = f(-x) = x^2. Conversely, a function is said to be odd if it is symmetric with respect to the origin, meaning that f(x) = -f(-x) for every value of x in the domain. For example, the function f(x) = x^3 is odd since f(x) = -f(-x) = -x^3.
6. Periodicity: A function is said to be periodic if it repeats its values at regular intervals. The smallest positive value of x for which f(x) = f(x + T) for some non-zero constant T is called the period of the function. For example, the sine function sin(x) has a period of 2π, since sin(x) = sin(x + 2π) for all real values of x.
Understanding these properties of functions can help us analyze and manipulate them more effectively in various mathematical applications.