WEBVTT
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In this question, we’ve gotta find the rule for the given function table.
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And then in the function table, we’ve got a row of 𝑥-values or input values and a row of output values.
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We can see that for an input of three, we get an output of twelve; for an input of six, we get an output of fifteen; and for an input of seven, we get an output of sixteen.
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Now in this question, we’ve been given some slightly random values for 𝑥.
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So we’ve got three and six and seven.
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Now the difference between three and six is three, but the difference between six and seven is only one.
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So it is not like we’ve got a common difference between our input values.
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So we’re gonna be a little bit careful about how we analyse this.
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Now looking at the corresponding output values, the difference between twelve and fifteen is three and the difference between fifteen and sixteen is one.
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Well that’s kinda of interesting because every time our input value increases by one, the corresponding output value also increases by one.
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So if we input- increase the input by three, the output will increase by three; if we increase the input by one, the output would increase by one.
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Now in these cases when the common difference in the input will generate exactly the same difference in the output, that tells us that the multiplier of 𝑥 in our function rule must be one.
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Now we’re not saying that the function rule is one 𝑥 — one times 𝑥.
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We’re saying that the function rule contains the term one times 𝑥.
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If the rule contains two times 𝑥, then every time I increase my input by one, my output would increase by two.
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If I had three times 𝑥, if I increase my input value by one, then the output value would increase by three, and so on.
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But the rule itself — one times 𝑥 — is not the right rule for this particular function table clearly because if I have an input value of three, one times three is three.
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But I’m getting twelve.
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If I have an input value of six, one times six is six.
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But I want fifteen.
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Now I have to add nine to three in order to get twelve.
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And I have to add nine to six to get fifteen.
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And also I have to add nine to seven to get sixteen.
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So in order to get the output value I’m looking for, I’ll take this one 𝑥 that we’ve generated — the three, the six, and the seven — and I have to add nine to it.
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So that’s my rule: one 𝑥 plus nine.
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Now typically if we got one times something, we don’t normally bother writing that one.
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So the most efficient way to write our answer is the function rule is 𝑥 plus nine.