Functions
This lesson is part of a series of lessons for the quantitative reasoning section of the GRE revised General Test. In this lesson, we will learn:
- Functions as equations
- Domain of Functions
Functions
An algebraic expression in one variable can be used to define a function of that variable. Functions are usually denoted by letters such as f, g and h.
For example, the algebraic expression 2x + 7 can be used to define a function f by
f(x) = 2x + 7
where f(x) is called the value of f at x and is obtained by substituting the value of x in the expression above.
For example, if x = 1 is substituted in the expression above,
the result is f (1) = 2(1) + 7 = 9.
It might be helpful to think of a function f as a machine that takes an input, which is a value of the variable x, and produces the corresponding output, f(x). For any function, each input x gives exactly one output f(x).
However, more than one value of x can give the same output f(x). For example, if g is the function defined by g(x) = x2 − 2x + 3 then and g(0) = 3 and g(2) = 3.
This video shows some examples of functions defined by equations.
Domain of Functions
The domain of a function is the set of all permissible inputs, that is, all permissible values of the variable x. For the functions f and g defined above, the domain is the set of all real numbers.
Sometimes the domain of the function is given explicitly and is restricted to a specific set of values of x. For example, we can define the function h by h(x) = x2 + 2 for −3 ≤ x ≤ 3*.* Without an explicit restriction, the domain is assumed to be the set of all values of x for which f(x) is a real number.
The following videos show how to find the domain of a function.
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