Shadowing Practice: Relations and Functions | Algebra - Learn English Speaking with 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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Vocabulary and speaking notes for this lesson
This speaking lesson is built on the video “Relations and Functions”. The speaker keeps coming back to these words: value, relation, function, negative, correspond. This video has 124 sentences and 1357 words to shadow. The speech runs for 12:25. The speaker talks slowly, about 109 words per minute, so there is time to copy every sound. 83% of the words are among the 3,000 most common in English; the rest is worth studying before you start.
Key vocabulary in this video
15 words from the video worth learning, with pronunciation and meaning:
| Word | Pronunciation | Meaning |
|---|---|---|
| relation noun | /ɹɪˈleɪʃ(ə)n/ | The manner in which two things may be associated. |
| correspond verb | /ˌkoɹəˈspɑnd/ | To be equivalent or similar in character, quantity, quality, origin, structure, function etc. |
| output noun | /ˈaʊtpʊt/ | Production; quantity produced, created, or completed. |
| input noun | /ˈɪnpʊt/ | The act or process of putting in; infusion. |
| domain noun | /ˌdəʊˈmeɪ̯n/ | A geographic area owned or controlled by a single person or organization. |
| vertical adjective | /ˈvɝ.tɪ.kəl/ | Standing, pointing, or moving straight up or down; parallel to the local direction of gravity; along the direction of a plumb line; perpendicular to something… |
| graph noun | /ɡɹæf/ | A data chart (graphical representation of data) intended to illustrate the relationship between a set (or sets) of numbers (quantities, measurements or… |
| associate noun | /əˈsoʊʃi.ət/ | A person united with another or others in an act, enterprise, or business; a partner or employee. |
| intersection noun | /ˈɪntəɹˌsɛkʃən/ | The junction of two (or more) paths, streets, highways, or other thoroughfares. |
| diagram noun | /ˈdaɪ.ə.ɡɹæm/ | A plan, drawing, sketch or outline to show the function or operation of something, or to show the relationships between the parts of a whole. |
| notify verb | /ˈnoʊtɪfaɪ/ | To give (someone) notice (of some event). |
| complicated adjective | /ˈkɑm.plɪˌkeɪ.tɪd/ | Difficult or convoluted. |
| confirm verb | /kənˈfɝm/ | To strengthen; to make firm or resolute. |
| examination noun | /ɪɡˌzæm.ɪˈneɪ.ʃən/ | The act of examining. |
| identical adjective | /ɪˈdɛntɪkl̩/ | Bearing full likeness by having precisely the same set of characteristics; indistinguishable. |
Phrasal verbs you will hear
| Word | Meaning |
|---|---|
| write out verb | To write at full length or in expanded form. |
| get back to verb | Return to (an activity) |
Pronunciation to watch
The speaker uses 23 contractions and reduced forms, such as we're, I'm, doesn't. Say them the short way, as you hear them.
- The “sh” and “zh” sounds: relation /ɹɪˈleɪʃ(ə)n/, associate /əˈsoʊʃi.ət/, intersection /ˈɪntəɹˌsɛkʃən/, examination /ɪɡˌzæm.ɪˈneɪ.ʃən/, preposition /ˌpɹɛpəˈzɪʃən/
- Long words — get the stress right: associate /əˈsoʊʃi.ət/, intersection /ˈɪntəɹˌsɛkʃən/, complicated /ˈkɑm.plɪˌkeɪ.tɪd/, examination /ɪɡˌzæm.ɪˈneɪ.ʃən/, preposition /ˌpɹɛpəˈzɪʃən/
How to practise with this video
- Listen to the whole video once without speaking and note the words you do not know.
- Shadow it sentence by sentence at normal speed, repeating each one until your rhythm matches the speaker.
- Record yourself and compare with the original, paying attention to words like relation, correspond, output.
What is the Shadowing Technique?
Shadowing is a science-backed language learning technique originally developed for professional interpreter training and popularized by polyglot Dr. Alexander Arguelles. The method is simple but powerful: you listen to native English audio and immediately repeat it out loud — like a shadow following the speaker with just a 1–2 second delay. Unlike passive listening or grammar drills, shadowing forces your brain and mouth muscles to simultaneously process and reproduce real speech patterns. Research shows it significantly improves pronunciation accuracy, intonation, rhythm, connected speech, listening comprehension, and speaking fluency — making it one of the most effective methods for IELTS Speaking preparation and real-world English communication.