# How to Find Domain and Range

## Domain and Range

Hello, and welcome to this video on domain and range! In this video, we will see

- What domain and range are
- And how to find the domain and range of a function

Remember, a function is a relation between two sets of numbers, an input and an output. Each element of the input produces a unique element of the output.

The domain of a function is the set of all possible inputs of a function. This means it is any number you can plug into a function. For most functions, this will be any number you can plug in for the letter x. Almost every time, your domain will be all real numbers, except for a few special cases like **square root** functions and rational numbers.

The range of a function is the set of all of the possible outputs of a function. Typically this will be represented by the letter y or f of x. The range is any number that you can get when you plug in any number for x.

Let’s look at a simple linear function: y equals 4x plus 3. y = 4x + 3. We are going to find the domain and range using just the equation, by looking at a graph, and by looking at a table.

Let’s think about this algebraically for a minute. The domain is any number we can put in place of the x. You could put 1, 2, negative 7, 84, or any other number in place of the x. This means that the domain is: negative infinity is less than or equal to x is less than or equal to positive infinity. Another way to say this is that the domain is the set of all real numbers.

What about our range? Well, if I plug in 1 for x, I get 7 and if I plug in 2 for x, I get 11. But I can also plug in 1.5 for x, which would give me 9, or 1.25 for x, which would give me 8. I can plug in any decimal number, so for this equation, I can also get out any number for y by searching for the right x. The range of this function is also the set of all real numbers.

Now I want to check this graphically. If we graph this function, we see that it is a line. Lines continue across every value of x and every value of y. This matches up with what we found out by thinking through it algebraically. This further proves that domain and range are both the set of all real numbers.

Now let’s look at a table of values for the first four terms of this function.

X | Y |

1 | 7 |

2 | 11 |

3 | 15 |

4 | 19 |

In this case, we are only looking at a portion of the function, so our domain of values would be {1, 2, 3, and 4} and our range of values would be {7, 11, 15, and 19}.

Let’s try a couple of examples. What is the domain and range of the function y equals x squared minus 4x plus 3? \(y=x^{2}-4x+3\)?

The domain is the list of numbers that can be plugged in for x. You can plug in any number for x, so the domain is the set of all real numbers.

What about our range? Let’s figure this out by looking at a graph of the equation.

Remember, our range is every possible value for y. If we look at our graph, we see that it is a **parabola** that opens up with a vertex at (2, negative 7) . This means that our range is y is greater than or equal to negative 7.

Let’s try one more example, this time using a table for the function y equals 2x minus 1 \(y = 2x – 1\).

X | Y |

7 | 13 |

14 | 27 |

21 | 41 |

28 | 55 |

What would be our domain and range given this table? Our domain would be {7, 14, 21, and 28} and our range would be {13, 27, 41, and 55}.

If we wanted the domain and range for the whole function, we would consider what numbers we can plug in for x and what corresponding y-values we would get. Well, we can plug in any number for x, and it is a linear function so we can get any number for y. Therefore, the domain and range of this function is all real numbers.

Remember, if you are finding the domain and range of a function algebraically, think about what numbers you can plug in for x and the resulting numbers you will get for y. If you are finding the domain and range given a graph, follow your finger along the graph and see what x-values it covers and what y-values it covers. Finally, if you are looking at a table, the domain is the list of numbers inputted for x and the range is the list of numbers that are the outputs of those x inputs, the numbers in the y column.

I hope this video on domain and range was helpful! Thanks for watching and happy studying!