Pratica di Shadowing: Relations and Functions | Algebra - Impara a parlare inglese con i video
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in this video we're going to focus on relations and functions
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so what is a relation a relation is a set of pairs of input
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and output values here
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we have three ordered pairs in the first relation on the
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left the x value is the input value the y value
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is the output value the x values is associated with the
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domain of the relation the y values is associated with the range of the relation
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so now let's focus on part a list the domain
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and range of each relation so let's start with the domain
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so what we're gonna do is we're gonna make a list of all of the x values
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and I'm going to write it in ascendant order.
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So first we have negative 3 and then 0 and 2.
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Now let's write the range of that relation.
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So we're going to focus on the y values and it's already listed in ascendant order.
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So 1, 4, and 5.
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Now, let's do the same thing for the other relation.
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So, let's write out the domain.
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So, the lowest x value is negative 2.
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Next is 1 and then 3.
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Now, let's write out the range of that relation.
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The lowest y value is negative 2.
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And then it's 3, 4, and 7.
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So that's how you can write out the domain and range of each relation.
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Now how can we determine if the relation is a function?
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In order for the relation to be a function, every input value must have only one output value.
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If an input value corresponds to two or more output values, that relation is not a function.
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Now, let's focus on the first relation.
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So we have the ordered pair 2 -1.
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The input value is 2, the output value is 1.
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And then negative 3, 4.
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So negative 3 corresponds to 4, and then 0 corresponds to 5.
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So for the first relation, we can see that for every input value, there's only one output value.
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Now let's focus on the second relation.
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So we have the ordered pair 1, 3.
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Next is negative 2, 4.
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And then it's 3, negative 2.
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And then finally, negative 2, 7.
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So for the second relation, notice that negative 2 corresponds to two different output values.
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Now that's a problem.
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If you put in an input value of negative 2, should the output be 4 or 7?
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So whenever you have that situation, you know that relation is not a function.
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The first one is a function.
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Every input value corresponds to an output value, just one output value.
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So a quick way to look at a relation to see if it's not going to be a function, look for repeated x values.
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If you see two x values that are the same but correspond to two different y values, then you know the relation is not a function.
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I want to take a minute to talk about my website video -tutor .net.
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It's a very simple website, not too complicated,
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but for those of you who want to be notified anytime I release specialized content in the form of a video,
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an ebook, an article.
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It could be a digital course or a podcast.
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If you want to be notified, feel free to join the email list.
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And once you confirm your email, you're going to get access to a page that has all of my playlists listed on it.
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And this includes my final exam videos and also my test prep videos.
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So feel free to join the email list when you get a chance.
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And let's get back to the video.
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Now let's move on to the next example.
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Draw a mapping diagram of each relation shown below.
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So let's start with the relation on the left.
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We're going to map out the domain and the range.
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So for the domain, we have the values negative 2, 1, and 3.
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For the range, we have the y values, negative 6, 0, and 4.
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Now, negative two corresponds to zero,
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one corresponds to four, three corresponds to negative six.
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So for every input value on the domain side, there's one corresponding output value on to range aside.
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So this is a relation.
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I mean this relation is a function.
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So the answer is yes for the first relation.
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Now let's move on to the second relation.
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So let's create a map and diagram as well.
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So let's start with the domain
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The lowest x value is negative 2
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Next we have 0
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And then the last one is 3
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Now looking at the y values
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The lowest one is negative 1, and then it's going to be 1, 2, and 5.
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So negative 2 corresponds to positive 1.
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0 corresponds to 5.
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3 corresponds to 2.
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and 0 corresponds to negative 1.
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So just by seeing the repeat x values that we see here, we could tell that this is not going to be a function.
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The two x values have two different y values.
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You can see 0 points to negative 1 and 5.
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So the second relation is not a function.
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Now for this one, what we're going to do is we're going to draw a function table of the relation.
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And then we're going to determine if the relation is a function.
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So in this table, we're going to list the input values next to the output values.
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The input values represent the x values.
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The output values represent the y values.
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so the input values it corresponds to the domain and the output values corresponds to the range
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so the lowest input value that we have is negative three next is one and then we have another one
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and then after that is it's three and five
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Now, for this function table, I'm going to write the input value twice, because that's what we have here.
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When writing out the domain and the range, for repeat values, we would write repeat values once.
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Now, negative 3 corresponds to 2.
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For the table, these numbers need to match, so I'm not going to list the output values in ascending order.
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Now, for this one, we could use either one.
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So, I'm going to use 1, 2 for the next one, and then 1, 4.
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Now, when x is 3, y is 7.
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And when x is 5, y is negative 4.
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So, that's the function table and because we have two identical x values that correspond to two different y values,
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we know that this relation is not a function.
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So that's it for this problem.
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When you have a graph, the best way to determine if the graph represents a function is to use the vertical line test.
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And that's what we're going to do in this problem.
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So any way you draw a vertical line for the first graph, notice that the line only touches the graph at one point.
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Therefore, the answer is yes, it represents a function.
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For the next one on the right, if we draw a vertical line, notice at this point, or placing a line at that location,
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we have three points of intersection between the graph and the vertical line.
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If we can get two or more points on a vertical line, then the relation is not a function.
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So we're going to say no. Now, if we put the vertical line here, notice that we have five points on that line.
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So this relation is not a function.
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For the next relation, it doesn't matter where we put the graph.
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We will only get, I mean, it doesn't matter where we put the vertical line we're only going to get one point.
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If we put it here, it's only going to touch the line once.
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So we can't draw a vertical line where it touches two points.
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Therefore, this relation represents a function.
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For the circle, if we put the line here, we can get two points of intersection.
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so we're going to say yes i mean no not yes
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this is no the circle does not represent a function
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it does not pass the vertical line test
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it touches the line at two points in order for it
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to pass the vertical line test the graph must touch the line only at one point
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as we saw in these two cases.
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So that's how you can use the vertical line test to determine if a relation represented by a graph is a function.
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Vocabolario e note di pronuncia per questa lezione
Questa lezione di conversazione si basa sul video “Relations and Functions”. Le parole che ritornano più spesso: value, relation, function, negative, correspond. Questo video contiene 124 frasi e 1357 parole da ripetere con lo shadowing. Il parlato dura 12:25. Chi parla lo fa lentamente, circa 109 parole al minuto, quindi c’è tempo per imitare ogni suono. Il 83% delle parole rientra nelle 3.000 più comuni dell’inglese; conviene studiare le altre prima di iniziare.
Vocaboli chiave di questo video
15 parole del video che vale la pena imparare, con pronuncia e significato:
| Parola | Pronuncia | Significato |
|---|---|---|
| relation sostantivo | /ɹɪˈleɪʃ(ə)n/ | parente |
| correspond verbo | /ˌkoɹəˈspɑnd/ | corrispondere |
| output sostantivo | /ˈaʊtpʊt/ | risultato, produzione |
| input sostantivo | /ˈɪnpʊt/ | immissione |
| domain sostantivo | /ˌdəʊˈmeɪ̯n/ | dominio |
| vertical aggettivo | /ˈvɝ.tɪ.kəl/ | verticale |
| graph sostantivo | /ɡɹæf/ | grafico |
| intersection sostantivo | /ˈɪntəɹˌsɛkʃən/ | incrocio |
| diagram sostantivo | /ˈdaɪ.ə.ɡɹæm/ | diagramma |
| notify verbo | /ˈnoʊtɪfaɪ/ | notificare |
| complicated aggettivo | /ˈkɑm.plɪˌkeɪ.tɪd/ | complicato, intricato |
| confirm verbo | /kənˈfɝm/ | cresimare |
| examination sostantivo | /ɪɡˌzæm.ɪˈneɪ.ʃən/ | esame, visita |
| identical aggettivo | /ɪˈdɛntɪkl̩/ | identico |
| anytime avverbio | in qualsiasi momento, in qualunque momento |
I phrasal verb che sentirai
| Parola | Significato |
|---|---|
| write out verbo | appuntare, emettere |
Pronuncia a cui fare attenzione
Chi parla usa 23 contrazioni e forme ridotte, come we're, I'm, doesn't. Pronunciale nella forma breve, così come le senti.
- I suoni “sh” e “zh”: relation /ɹɪˈleɪʃ(ə)n/, associate /əˈsoʊʃi.ət/, intersection /ˈɪntəɹˌsɛkʃən/, examination /ɪɡˌzæm.ɪˈneɪ.ʃən/, preposition /ˌpɹɛpəˈzɪʃən/
- Parole lunghe — attenzione all’accento: associate /əˈsoʊʃi.ət/, intersection /ˈɪntəɹˌsɛkʃən/, complicated /ˈkɑm.plɪˌkeɪ.tɪd/, examination /ɪɡˌzæm.ɪˈneɪ.ʃən/, preposition /ˌpɹɛpəˈzɪʃən/
I suoni difficili per chi parla italiano:
- Consonante finale — senza aggiungere una vocale dopo: correspond /ˌkoɹəˈspɑnd/, output /ˈaʊtpʊt/, input /ˈɪnpʊt/, graph /ɡɹæf/, associate /əˈsoʊʃi.ət/
- /æ/ — più aperta della “e”: graph /ɡɹæf/, diagram /ˈdaɪ.ə.ɡɹæm/, examination /ɪɡˌzæm.ɪˈneɪ.ʃən/, podcast /ˈpɑdˌkæst/
Come esercitarsi con questo video
- Ascolta tutto il video una volta senza parlare e annota le parole che non conosci.
- Fai shadowing frase per frase a velocità normale, ripetendo ognuna finché il tuo ritmo coincide con quello di chi parla.
- Registrati e confronta con l’originale, facendo attenzione a parole come relation, correspond, output.
Cos'è la tecnica dello Shadowing?
Shadowing è una tecnica di apprendimento delle lingue supportata da studi scientifici, originariamente sviluppata per la formazione dei traduttori professionisti e resa popolare dal poliglotta Dr. Alexander Arguelles. Il metodo è semplice ma potente: ascolti un audio in inglese di madrelingua e lo ripeti immediatamente ad alta voce — come un'ombra che segue il parlante con un ritardo di solo 1–2 secondi. A differenza dell'ascolto passivo o degli esercizi di grammatica, lo shadowing costringe il tuo cervello e i muscoli della bocca a elaborare e riprodurre simultaneamente i modelli di discorso reale. La ricerca dimostra che migliora significativamente la precisione della pronuncia, l'intonazione, il ritmo, il discorso connesso, la comprensione dell'ascolto e la fluidità del parlato — rendendolo uno dei metodi più efficaci per la preparazione alla prova di speaking dell'IELTS e per la comunicazione reale in inglese.