In this video, we are going to discuss the language of relation and functions. Relations abound in daily life. People are related to each other in many ways, as parents and children, teachers and students, employers and employees, and many others.
In business, things that are both are related to their costs and the amount paid is related to the number of things both. When we say relation, it is a rule that relates values from a set of values, we call that as a domain, to a second set of values and we call that as a range. The elements of the domain can be imagined as input to a machine that applies a rule to these inputs to generate one or more outputs. A relation is also a set of ordered values.
pair x and y. For example, a relation R have a set of ordered pair 1, 2, 2, 4, 3, 6, 4, 8, 5, 10. So, yung 1, 2, 3, 4, and 5 ay tinatawag nating domain. Yung 2, 4, 6, 8, and 10 ay tinatawag naman nating range. Okay, a relation is a set as a subset. So, for example, let set A is equal to 1 and 2 and set B is equal to 1, 2, 3 and define a relation R from A to B as follows.
So, given the statement that your X and Y is an element of the product of set A and B, b. So x and y is an element of relation r. It means that, so by this statement, x minus y over 2 is an integer.
So we're going to answer the following questions. State explicitly which ordered pairs in A times B and which are in relation R. Is 1 related to 3? Is 2 related to 3? Is 2 related to 2?
1. What are the domain and range of relation R? Okay, let's answer number one question. So first, we need to get the product of set A and B.
So to get the product, so that is... 1, 1, so ito yan, 1, 1, 1, 2, 1, 3, 2, 1, 2, 2, and 2, 3. So, all of this, this set of ordered pairs, isa-isahin natin isa-substitute on the given statement. So, kailangan maging element sila ng relation R kapag ang output nito is an integer. So, kapag hinti. Hindi sila element ng relation R.
So, unahin natin yung 1 and 1. 1 and 1 is an element of relation R. Why? So, kapag sinubstitute natin on the given statement na kung saan ay dapat integer ang sagot natin. So, pag sinabi natin integers, so, yan ay included dyan yung negative numbers, zero and positive numbers.
Since the... Output is 0 and 0 is an integer. So, therefore, 1 and 1 is an element of relation R.
Next, 1, 2. 1, 2 is not an element of relation R. Why? Kapag sin 1 half, negative 1 half fraction ang makukuha nyo.
So, this is not an integer. 1, 3 is an element of relation R. Why? So, kapag sinaptitude natin on the given statement, makukuha natin negative 1. And negative 1 is an integer. So, ibig sabihin, 1 and 3 is an element of...
relation are next two and one is not an element of relation are why so I know my one nothing capacity to cheat nothing young two and one saw given statement so my one not in one hop and one hop is not an integer another two and two is an element of relation are why so my one nothing Jen zero zero is again is an integer next 2, 3 is not an element of relation R because negative 1 half is not an integer. Next. So, therefore, ano lang yung mga element na in relation R?
So that is 1, 1, 1, 3, and 2, 2. Number 2 question, is 1 related to 3? Is 2 related to 3? Is 2 related to 2? So given yung sagot natin kanina, so dito natin pagbabasihan yung sagot natin sa question number 2. So is 1 related to 3?
Yes, because 1 and 3 is an element of relation R. And then 2 is related to 3? The answer is no because 2 and 3 is not an element of relation R. Saan yan?
Ito yan. Because negative 1 half yung output natin dyan. And negative 1 half is not an integer. Next, is 2 related to 2?
The answer is yes because 2 and 2 is an element of relation R. Question number three, what are the domain and range of R? So in the given problem, set A is our X values, right? And set B is our Y values. So therefore, the domain for relation R is 1 and 2, and the range for this given, so that is our B, that is 1, 2, and 3. Function.
When you say function, it is a relation where each element in the domain is related to only one value in the range by sum rule. Next, it is the element of the domain can be imagined as input to a machine that applies a rule so that each input corresponds to the same value. to only one output.
So, kanina, nung dinify natin si relation, so, yung output nya, it's, pwede maging one or more output. Pero kapag function, dapat meron lang syang only one output. Another, a function is a set of ordered pairs, x and y, such that no two ordered pairs have the same x value but different y values. So, we will remember this.
There should be no two ordered pairs that have the same x values. And they should have different y values or their output should be different. Okay.
Function can be represented in different ways. So, kagaya ng table of values, ordered pairs, graph, and equation. So, may kita natin. So, we can represent function in these four different ways. So, which of the following relation are function?
First, relation F is equal to 1, 2, 2, 2, 3, 5, 4, 5. So, is that a function? Yes, it is a function. Why? We don't have an X value that is repeated. So, in relation G, we have 1, 3, 1, 4, 2, 5, 2, 6, and 3, 7. So...
Is it a function? This is not a function. Why? We have two x values that are repeated.
So, we will remember, if the ordered pair is given, it should be just one x value. Okay? Next, in relation H, we have 1, 3, 2, 6, 3, 9 up to n, 3n.
So, in this given, is it a function or not a function? Of course, that is a function. So, if we have a value here, we will not get an X value that is repeated. So, therefore, relation H is a function.
Okay, also function can be represented using mapping diagrams. So, in the first mapping diagram, so tingnan natin, our x values ay merong unique output. Ibig sabi, sabihin one to one sila. So, sa isang x values, merong isang y value. So, ngayon, that is a function.
So, tatandaan natin kapag one to one using mapping diagram, that is a function. Next. Okay, meron tayong 5, 7 na input or x value na may isang output lang, that is 1. At meron naman tayong 6, 8, and 9 na may isang output lang din.
Therefore, ang tawag naman natin dito is, to 1. Kapag main to 1, that is also a function. Okay? Tatandaan natin kapag 1 is to 1 or main to 1, that is a function. Okay, in the third mapping diagram, meron tayong 7, correspond sya sa dalawang y value natin. Ibig sabihin, si 7, meron 11 and 12. 13. So, ang tawag naman natin dito ay one to many.
So, kapag one to many, class, that is not a function. Tatandaan yan. Ulitin ko lang.
Kapag one to one, that is a function. Many to one, that is also a function. But, if that is one to many, that is not a function.
The vertical line test, this is a graph represent a function if and only if each vertical line intersect the graph at most once. So, ginagamit natin sa vertical line test para ma-identify natin kung ang graph ba ay function or not a function. So, kapag ginuhitan mo yun ang vertical line at isang point lang doon sa graph, ang madadaanan nya, yan ay function.
Pero kapag... Ginuhitan mo yun ng vertical line test sa kahit anong parte ng graph at meron kang madadaanan ng two or more point dun sa graph, yan ay not a function. So, for example, in this graph, okay, so kapag ginuhitan natin to ng vertical line test kahit sa ang parte ng graph dyan, isang point lang ang dadaanan natin. Therefore, yan ay function.
Next. Dito rin, kahit saan mo guhitan dyan ng vertical line test, isang point lang ng graph ang tatamaan mo So, function din yan Next Also, yung pangatlong graph natin, kahit saang party yan, o saang party ng graph, guhitan mo ng vertical line Ibig sabihin, isang point lang ng graph na yan ang tatamaan mo o madadaanan mo Kaya, function din yan Next Next Okay, in this example, let's look at it. For example, let's draw a vertical line here. So, it means this is not a function because these two points of the graph, this one and this one at the bottom, are what the vertical line test did.
So, it means this is not a function. It should be just one point of the graph to know that it is a function. So therefore, this is not a function.
This one. This is also not a function because if you test it using the vertical line, we will pass through two points of the graph. Okay, using the equation, we can identify the given equation if that is a function or not. Bakit?
Okay, paano ito? This is an example of linear function. So kapag linear, diba, ang graph niya ay... vertical straight line So, kapag ginamitan natin ng vertical line test yung graph ng linear function which is straight line we will only hit one point in the graph. So, remember, if it's linear, okay, in the degree of 1, that's a function.
Also, if it's a quadratic function, if it's a quadratic function, we will remember, linear function, quadratic function, is a function also. Why? The graph of the quadratic function is a parabola. So, if the parabola is like this, so, if... You did a vertical line test, you will not pass more than 2 points of graph.
You will remember, if it's linear and quadratic, it's a function. This one, so this is not a function. Remember, if your y is squared, this. So, kapag may squared y, that is not a function. Bakit?
So, kapag nilipat natin sa x squared dun sa kabila, magkakaroon tayo ng dalawang value ng y. Okay. Next, okay, y is equal to square root of x plus 1. This is also a function.
Next, y is equal to 2x plus 1 over x minus 1, also a function. Okay, kapag green up natin ito, makikita nyo na walang dalawa o mahigit pang sa dalawa ang dadalawang function. sa kalaanan ng vertical line test.
So, ito lang yung tatandaan nyo sa number 3. Kapag may squared yung y nyo. So, automatic that is not a function. Next.
In evaluating a function, so, for example, Q of x is equal to x squared minus 2x plus 2. When your x is equal to 2. So, ang gagawin lang natin, papalitan nyo lang yung lahat ng x nyo ng 2. Okay, we will replace 2. So, it will be 2 squared minus 2 times 2. So, we removed x, we replaced 2. We removed x there, we replaced 2. And then simplify. 2 squared is 4. Negative 2 times 2, that is negative 4 plus 2. 4 minus 4, that is 0 plus 2. So, it means q of 2 is equal to 2. Another. We have f of x is equal to 2x plus 1 when your x is 3x minus 1. So what does this mean? You will just replace the x with 3x minus 1. So f of 3x minus 1 is equal to 2x plus 1. So we will replace the x with 3x minus 1. And then distribute the 2 inside the parenthesis.
So 2 times 3x, that is 6x. 2 times negative 1, that is negative 2. plus 1. So, and then simplify, combine similar terms. Since wala namang kasamang may one variable tong 6x natin, so just copy. Then negative 2 plus 1, the answer is negative 1. Thank you for watching this video.
I hope you learned something. Don't forget to like. subscribe, and hit the bell button so you're updated for more video tutorials.
This is your guide in learning your math lesson, your Wow Math channel.