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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.

Rotate to landscape screen format on a mobile phone or small tablet to use the **Mathway** widget, a free math problem solver that **answers your questions with step-by-step explanations**.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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