Hi there. This math post is about domain and range of functions. This topic is seen in high school pre-calculus and calculus courses. It is also seen in introductory calculus courses in universities.

Math text rendered with the use of LaTeX (lay-tech) and QuickLaTeX.com.

## Topics

- Quick Review Of Independent & Dependent Variables
- What Is Domain?
- What Is Range?
- Domain and Range Of Various Functions Examples

## Quick Review Of Independent & Dependent Variables

An independent variable is an unknown quantity that does not depend on another variable for its amount. One common example of an independent variable is time denoted by `t`

. In most cases, the independent variable is denoted by the letter `x`

.

A dependent variable is a variable whose value depends on the independent variable. It is common to see the letter `y`

for representing a dependent variable. An example of a dependent variable is a worker's take home pay as it is dependent on the number of hours worked.

## What Is Domain?

In a mathematical function, the domain of a function is the set (collection) of all permissable values for the independent variable. You can also view the domain of a function as a collection of valid inputs for the function.

Given a function `f(x)`

, the domain of `f(x)`

is the set of all possible x values for the function.

**Example One**

For a simple linear function such as g(x) = x + 1, the domain is any x-value (x belong to the real numbers). There is no restriction for the values of x that can be inputted into the function.

**Example Two - Quadratic Function**

With a quadratic function such as , the domain is also any x-value (x belong to the real numbers).

**Example Three - Square Root**

This example features a function that has a domain that is not all real numbers for the independent variable.

It is known that it is not possible to take the square root of a negative number (in the real numbers). For the function , the domain is . Values for x can be 0 or greater (non-negative).

Different types of functions will have their own domain and ranges.

## What Is Range?

A function's range refers to the possible outputs of a function given its function domain. The range is associated with the dependent variable of a function.

**Example One**

Let's revisit the linear function g(x) = x + 1. The range for this function is g(x) belongs to all real numbers . The dependent variable can take on any value given a value from x.

**Example Two**

With a quadratic function, you have to look for whether the parabola opens up or down from the sign in front of the x-squared term.

Given the function the parabola opens upwards and its minimum is at point (0, 4). The range for this parabola function is and y belongs to the real numbers.

If given a different parabola function, you may need to use other techniques such as factoring quadratics or completing the square in order to determine the parabola vertex and intercepts.

**Example Three - Rational Function**

The rational function is of the form:

where the domain has a restriction of .

The range for the rational function is and y belongs to the real numbers. There is no x-value that can make y equal to 0.

## Domain and Range Of Various Functions Examples

**Example One**

The downwards facing parabola has a vertex that is a maximum at (0, -1). The domain for this function is x belongs to all real numbers. Since the vertex is a maximum, the range for this parabola is y is less than or equal to -1 with y belonging to the real numbers ().

**Example Two**

The trigonometric function has a no restrictions on its domain. Any numeric value or angle can be used for its input. With the sine function, the value of sine is between -1 and +1 inclusive. The domain for the sine function is with .

**Example Three**

Consider the simple exponential function . There is no restriction on its domain so it is . For the range, it is important to note the exponential function has no x-intercept as there is no exponent that makes two to the power of x equal to zero. The exponential function is greater than 0 and has no upper limit. The range here would be f(x) > 0.

**Example Four**

What if you have a function that is a combination of different function types? This example deals with a mixture of a reciprocal function and a square root function.

What is the domain and range for .

Let's look at the square root function and the reciprocal function separately.

With the square root function, x has to be at least 0 and the square root is a number that is at least zero in its range.

With the reciprocal function , the domain is and the range is .

For , the domain is what is common between the square root function domain and the reciprocal function domain. The domain for g(x) is x > 0. With the range, it is g(x) > 0.

The plot of this function looks like this.