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