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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

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 2*x* + 7 can be used to
define a function *f* by

*f*(*x*) = 2*x* + 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) = *x*^{2} − 2*x* + 3 then and *g*(0) = 3 and *g*(2) = 3.

This video shows some examples of functions defined by equations.

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*) = *x*^{2} + 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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