Functions versus relations

There are many ways of viewing at functions. We will look into a few. But initially, we need to talk about a few terms.

A "relation" is merely a relationship between sets of data. For example, all the people in one of your classes have different heights. The pairing of their names and heights is a relation.

In relations and functions, the pairs of names and heights are "ordered", which means one comes first and the other follows as second.

In order words, we could set up this pairing so that either you provide me with a name, and after that I give you that person's height, or rather you would give me a height, and I would give you the names of all the people who are as tall as that height.

The set of all the starting points is referred to as "the domain" and the set of all the concluding points is referred to as "the range." The domain is what you begin with; the range is what you end up with. The domain is the x's; the range is the y's.

A function is a "well-behaved" relation. Just like with members of your own family, a few members of the family of pairing relationships are better behaved than other. What this implies is that whilst all functions are relations, since they pair information, not all relations are functions.

Functions are a sub-division of relations. When we say that a function is "a well-behaved relation", we mean that, with the given a starting point, we know precisely where to go; given an x, we get just and precisely one y.

If we go back to the relation of your classmates and their heights, and let's assume that the domain is the set of everybody's heights. Let's assume that there's a fast food-delivery guy waiting in the hallway.

And all the delivery guy is aware of is that the food is meant the student in your classroom who is five-foot-five. Now allow the guy to come in. Who does he go to? What if nobody is five-foot-five?

What if there are six people in the rooms that are five-five? Will they all have to pay? What if you are five-foot-five? And what if you're bankrupt and as well allergic to the fast food he is bringing in?

The relation "height shows name" is not well-behaved. It is not a function. Given the relationship (x, y) = (five-foot-five person, name), there might be six various possibilities for y = "name".

For a relation to be a function there ought to be only and exactly one y that corresponds to a given x.

The "Vertical Line Test"

Viewing at this function graphically, what if we had the relation that is made up of a set of just two points: {(2, 3), (2, –2)}? We already know that this is not a function, since x = 2 goes to each of y = 3 and y = –2.

This characteristic of non-functions was observed by I-don't-know-who, and was codified in "The Vertical Line Test": Given the graph of a relation, if you can draw a vertical line that crosses the graph in more than one place, in that case the relation is not a function. Below are a few examples:

This graph illustrates a function, because there is no vertical line that will cross this graph two times.

This graph does not illustrate a function, due to the fact that any number of vertical lines will cross this oval twice. For instance, the y-axis crosses the line twice.

Deciphering is a statement is a function without the graph

Consider the entire graph you have created so far. The easiest method is to solve for "y =", Create a T-chart, select a few values for x, solve for the corresponding values of y, plot your points, and link the dots.

This is not essential for graphing, but the method offers you another way of discovering functions: If you can solve for "y =", then it's a function. In other words, if you can enter y into your graphing calculator, then it's a function. The calculator can only take care of functions. For instance, 2y + 3x = 6 is a function, because you can solve for y:

2y + 3x = 6

2y = –3x + 6

y = (–3/2)x + 3

On the contrary, y2 + 3x = 6 is not a function, due to the fact that you cannot solve for a unique y:

What It means is that this is solved for "y =", but it's not unique. You have to wonder if you are supposed to take the positive square root, or the negative square root. Apart from that, you couldn't locate the key "±" on your graphing calculator? So, in this situation, the relation is not a function.

(You can as well figure this out by making use of our first definition from above. Think of "x = –1". Then we get y2 – 3 = 6, so y2 = 9, and then y can be either –3 or +3. That is, if we created an arrow chart, there would be two arrows coming from x = –1.)

Functions: Domain and Range

Let's go back to the subject of domains and ranges. When functions are first introduced to you, you will likely be given a few simple "functions" and relations to deal with, being merely sets of points. These won't be very useful or exciting functions and relations, but your test is just meant to give you an idea of what the domain and range of a function are.

For example:

State the domain and range of the function below: Is the relation a function?

{(2, –3), (4, 6), (3, –1), (6, 6), (2, 3)}

The above list of points, because they are a relationship between definite x's and certain y's, is a relation. The domain is all the x-values, and the range is all the y-values. To produce the domain and the range, we merely list the values without duplication:

Domain: {2, 3, 4, 6}

Range: {–3, –1, 3, 6}

It is conventional to list these values in numerical order, but it is not needed. Sets are known as "unordered lists", so you can list the numbers in any order you want. You must be careful not to duplicate, though. Technically, duplications are okay in sets, but the majority of instructors deduct mark for you for doing that.

While the given set does represent a relation (because x's and y's are being related to each other), they would give you points with the same x-value: (2, –3) and (2, 3). Since x = 2 gives two possible destinations, then this relation is not a function.

Observe that the only thing you need to do to check if the relation was a function was to look for duplicate x-values. If you find a repeated x-value, then the various y-values mean that what you have is not a function.

State the domain and range of the relation below: Is the relation a function?

{(–3, 5), (–2, 5), (–1, 5), (0, 5), (1, 5), (2, 5)}

You would merely list the x-values for the domain and the y-values for the range:

Domain: {–3, –2, –1, 0, 1, 2}

Range: {5}

This is another example of a "dull" function, like the previous example.

Every last x-value goes exactly to the same y-value. But every x-value is different, therefore, while dull, this relation is really a function. In point of fact, these points lie on the horizontal line y = 5.

There is one other situation for finding the domain and range of functions. You will be given a function and asked to find the domain (and possibly the range, too).

Determine the domain and range of the given function:

The domain is all the values that x is permitted to take on. The only problem you would have with this function is that you ought to be careful not to divide by zero. So the only values that x cannot take on are those which would cause division by zero. Therefore, set the denominator equal to zero and solve; the domain will be everything else.

x2 – x – 2 = 0

(x – 2)(x + 1) = 0

x = 2 or x = –1

Then the domain is "all x not equal to –1 or 2".

From the picture, the graph "covers" all y-values. That is, the graph will go as low as possible and as well go as high as possible. Since the graph will finally cover all possible values of y, then the range is "all real numbers".

Determine the domain and range of the given function:

The domain is all possible values of x. The only issue with this function is that you cannot be able to obtain a negative inside the square root. Therefore, you ought to set the insides greater-than-or-equal-to zero, and solve. The result will be your domain:

–2x + 3 > 0

–2x > –3

2x < 3

x < 3/2 = 1.5

Then the domain is "all x > 3/2".  

The range needs a graph. You ought to be cautious when plotting a graph of radicals:

The graph starts at y = 0 and goes down from there. Whilst the graph goes down very slowly, ultimately, It will let you to go as low as possible by choosing picking an x that is adequately large. Again, the graph will never start coming back up judging from graphing experience. Then the range is "y < 0". View more on psalmfresh.blogspot.com for more educating and tricks article Thank you