Did you know that the length of the shadow of an object is related to its height in a very special way, and we can find this interesting relationship using functions?

**Here are a few examples of the real-life applications of a function.**

Real-life applications | Is related to (is a function of) |
---|---|

The perimeter of a circular arena | Diameter |

Temperature | Number of factors and inputs |

Bodyweight of an individual | Age and height |

The position of a moving object | Time |

Simple interest | Time, principal, and the interest rate |

Let’s learn about domain definition, domain definition math, range definition, domain calculator, domain definition.

Check out the interactive simulation to know more about the lesson and try your hand at solving a few interesting practice questions at the end of the page.

**Lesson Plan**

**What is Domain?**

A domain is ‘all the values’ that go into a function.

The domain of a function is the set of all possible inputs for the function.

Before we proceed, let us know what a function is.

**What is a Function?**

A function is something that receives an input to produce a definite output.

Consider this box as a function.

\[ \text {Function } = \text { f}(\text {x}) \]

and here \[ \text { f}(\text {x}) = (\text {x})^2 \]

Functions assigns a unique output to each value that has been input.

Example:

When the function \( \text { f} (\text {x})= \text { x }^2 \) is given, the values \( \text { x } = {1,2,3,4...\text {n},} \) then the domain is simply those values \( { 1,2,3,....\text {n}, } \) and the output values are called the range.

\( \text {Domain} → \text {Function} → \text {Range} \) |

**How is a Domain Related to a Function?**

The domain of a function is the set of all possible inputs for the function.

Input (Domain) | Function (Squaring) | Output (Range) |
---|---|---|

x = \(1, 2, 3,... \) | \(\text { f} (\text {x}) = \text {x}^2 \) | \(1, 4, 9,..\) |

y = \(..-3, -2, -1,0,1,2, 3,..\) | \(\text { g} (\text {y}) = \text {y}^2 \) | \(0,1, 4, 9,..\) |

**Note:** Both functions square the input, but they have a different set of inputs, and thus give a different set of outputs.

The range of \( \text {g} (\text {y})\) also includes \(0\)

\( \text {f} (\text {x}) \) always gives a unique answer, but \( \text {g} (\text {x})\) can give the same answer with two different inputs such as \( \text {g} (-3)=9 \), and also \( \text {g} (3)=9 \)

Want to understand the “Why” behind the “What”? Explore Domain Definition with our Math Experts in Cuemath’s LIVE, personalized, and interactive online classes.

The range is a subset of codomain.

**The Importance of Codomain**

Remember: A function must be single-valued. It cannot give back two or more results for the same input.

For a function \[ \text { f} ( \text {x}) = \sqrt{ \text { x} } \]

In the case of a codomain (the possible outputs) of the set of real numbers is \[ \text { f} (4) = 2 \text { or } -2 \]

So \( \text { f} ( 4) = 2 \text { or } -2 \, \) is not right!

Now, the question arises: Can the square root be called a function?

It can, but only after a certain modification. This can be fixed by simply putting a limit to the codomain to non-negative real numbers.

So \( \text { x }= 2 \) only.

Hence, an incorrect codomain can actually affect whether something is a function or not.

**How to Find the Domain of a Function Using Algebra?**

We can determine the domain of a function either algebraically or by graphical method.

To calculate the domain of a function algebraically, you simply solve the equation to determine the values of \( \text { x }\).

- A function relates each element of a set with exactly one element of another set (possibly the same set).
- The domain of a function f is all of the values for which the function is defined.
- To find the domain of a function f, you must find the values for which f is not defined.

**How to Find the Domain of a Function Using a Graph?**

Another way to identify the domain and range of functions is by using graphs.

A domain refers to the set of possible input values, the domain of a graph consists of all the input values are shown on the \( \text { x}-\text {axis}\).

The range is the set of possible output values, which are shown on the \( \text { y}-\text {axis} \).

**Let us see the domain and the range of some different types of functions:**

- Can you describe a function whose domain and range are the same?

**Solved Examples**

Example 1 |

Find the domain of the function:

\( { (-3,4), (0,0), (3,-4), (12, -8), (16,-12) } \)

**Solution**

Domain is the set of the first coordinate of the ordered pair.

\( \therefore\text {Domain } = \{ -3, 0, 3, 12, 16 \} \) |

Example 2 |

What is the domain of the function \( \text { f} (\text {x}) = ( \text {x}^2 - 1) \) ?

**Solution**

Here, the given function has a variable \( \text {x} \) which is squared, and then the result is lowered by \( 1 \)

We know, any real number can be squared and then lowered by \( 1 \)

Hence,

The domain is the set of real numbers without any restrictions.

\(\therefore \text{Domain} \, \text { of } \,\text { f } = (-\infty , \infty)\) |

Example 3 |

Find the domain of the function \( \text { f (x)}= \dfrac { \text {x}+1}{3-\text {x}}. \)

**Solution**

At first, we will set the denominator equal to \( 0 \), and then we will solve for \( \text {x} \).

\[ \begin{align*} 3 - \text {x} &= 0 \\ -\text {x} &= -3 \\ \text {x} &= 3 \end{align*} \]

Hence, we will exclude \( 3 \) from the domain.

Domain : All real numbers where \( ( \text {x} < 3) \) and \( (\text {x} > 3 ) \)

For combining two sets we use "\( \cup \)" symbol.

\(\therefore \text{Domain} \, \text{ of } \, \text{ f } = (-\infty,3) \cup (3, \infty) \) |

Example 4 |

What is the domain of the function \( \text { f} (\text {x} )= \sqrt{ (6-\text {x})} \)?

**Solution**

Here we exclude any real number that results in a negative value in the radicand.

\(\therefore\) radicand \[ \begin{align*}( 6 - \text {x} ) &\geq 0 \\ \\ - \text {x} &\geq -6 \end{align*} \]

Hence, all the numbers greater than \( 6 \) will be excluded from the domain.

\(\therefore \text{Domain} = ( -\infty, 6 ]\) |

Example 5 |

Find the range and domain of the function given below:

\[ \text { f} ( \text {x}) = \text {x}^2 \]

**Solution**

We observe from the graph that the horizontal extent of the graph is from \( ( -\infty, \infty) \).

So, the domain is \( ( -\infty, \infty) \).

Now the vertical extent of the graph goes from \( 0 \) to \( \infty \),

So, the range is \( ( 0, \infty) \).

Since for a quadratic function \( \text { f} (\text {x}) = \text {x}^2 \), domain is all real numbers.

And the range is only non-negative real numbers.

\( \therefore \begin{align*} \text{Domain} &= (-\infty, \infty) \\ \text{Range} \, &= \, [ 0,\infty) \end{align*} \) |

**Interactive Questions**

**Here are a few activities for you to practice. **

**Select/Type your answer and click the "Check Answer" button to see the result.**

**Let's Summarize**

We hope you enjoyed learning about domain definition with the simulations and practice questions. Now you will be able to easily solve problems on domain definition math, range definition, domain calculator, domain definition**.**

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**Frequently Asked Questions (FAQs)**

## 1. What is another name for domain in math?

In mathematics, the domain is sometimes referred to as the domain of a function.

The domain of a function is the complete set of possible values of the independent variable.

## 2. How do you write the domain and the range?

Domain and range are written in set builder notation.

For example, \( \{ {\text x \, \epsilon \,Z \, | \,\text x ≥ 5 \text { and x} ≤ 9 }\} \)

It starts with all real numbers, then it limits them between 5 and 9 inclusive.

## 3. Can a function have an empty domain?

No, a function cannot have an empty set as the domain or empty domain.

The domain is valid only if it contains some value in it.