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Domain and path of a function by carlos84

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· @carlos84 ·
$2.89
Domain and path of a function
<div class="text-justify">


<center>![image.png](https://images.hive.blog/DQmR7nWmWRnhQ5qzAQG5Da7LMBBYGMR39ijpKiupaPVXrEt/image.png)</center>


The domain of a function can be written either explicitly or implicitly through the equation used to define the function. The implicit domain is the set of all real numbers for which the equation is defined, while an explicitly defined domain is the one given together with the function. For example, the function given by:


![image.png](https://images.hive.blog/DQmcWy5Acdj1fYUDBKPtpGGoCRMbnfNQCrmSupumpjoxMfD/image.png)

As can be seen, the function has an explicit domain, since the function is only defined for the***x*** that are between 4 and 5.

On the other hand, if we express the function as follows:

![image.png](https://images.hive.blog/DQmNxSPBvuo18tSmib6947Zbw8FL6Rosm5G3Wgvj5k3eqeS/image.png)
 
The domain for this case is expressed implicitly, i.e. the domain is defined for the values of x for which the function exists, and for this we must perform a general analysis to see which are the values of x for which the function exists.  

For the case of the rational function, it exists for any value of x, except for x = 2, since when substituting the 2 and -2 in x<sup>2</sup> gives us 4, and then subtracting with 4 gives us zero, which implies that division by zero is not defined. 

The way to express this domain implicitly is:

The domain is the set:   

# {š‘„: š‘„ ≠ ±2}

Which means that the domain is all real numbers except 2 and -2.

### ***Calculation of the domain and path of a function***

<p>

To calculate the domain and path of a real function, I am going to propose to find the domain and path of the following function:

<center>![image.png](https://images.hive.blog/DQmc14GGEf6ovHwDQprGYo2ZhUMERVv67CPgtsr6vn7YoC6/image.png)</center>

The first thing we must take into account is that what is inside a square root must be greater than or equal to zero, which implies that:

<center>![image.png](https://images.hive.blog/DQmQ9yn7VohFXXEdRvdD3KTDCo8SXhrpT9g4Vs8tgKatdES/image.png)</center>

We clear the variable x:

<center>![image.png](https://images.hive.blog/DQmaCBz3zutss68qeGiu499LAE4LaAnZQEEaZhv6ZBAiiD2/image.png)</center>

That x is greater than or equal to one, means that x is the set of values that have to be greater than or equal to 1, i.e. the domain is the set that spans the following interval [1;āˆž)

The graph of the domain is:

![image.png](https://images.hive.blog/DQmQzC63fr7FKb2umYrSpNJtw8cxyRBt3rDxCb89U4biN8W/image.png)

To find the path or range of the function we observe that the images y = f(x) are never negative, which means that the range or path is the set [0;āˆž), i.e. the graph on the Y-axis goes from zero to positive infinity.

### ***Bibliographic Reference***

<p>

Calculus with Analytic Geometry by Ron Larson, Robert, P. Hostetler and Bruce H. Edwards. Volume I. Eighth Edition. McGraw Hill. AƱo 2006 

</div>
šŸ‘  , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , and 220 others
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vote details (284)
@stefano.massari ·
Great!!! Thanks for this post that made me review the theory of functions, I had forgotten some terms. so I also reviewed the concept of domain. I had forgotten that the domain of a function is precisely the set of all input values ​​(i.e. the x) for which the function is defined, i.e. for which it makes sense to calculate the output (i.e. the y or f(x)). It is impossible to remember everything!
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@carlos84 ·
$0.11
Hello friend, certainly this concept of domain is very important, since later when we see the concept of derivative of a function will be paramount. Greetings and thanks for your comment.
šŸ‘  
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