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Question Video: Graphs of Cube Root Functions Mathematics • 10th Grade

Which of the following is the graph of ๐‘“(๐‘ฅ) = โˆ›๐‘ฅ โˆ’1? [A] Graph a [B] Graph b [C] Graph c [D] Graph d

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Video Transcript

Which of the following is the graph of ๐‘“ of ๐‘ฅ equals the cube root of ๐‘ฅ minus one?

There are several different ways in which we can approach a question like this. One way is to take several different input values or ๐‘ฅ-values and see what their corresponding output or ๐‘“ of ๐‘ฅ values would be. Letโ€™s say that we take the values of ๐‘ฅ of negative one, zero, and one. When ๐‘ฅ is equal to negative one, the cube root of negative one is negative one. And then subtracting one from that will give us negative two. When ๐‘ฅ is equal to zero, the cube root of zero is zero, and subtracting one will give us negative one. Finally, when ๐‘ฅ is equal to one, the cube root of one is one, and subtracting one will give us an output of zero.

We then know that the graph must contain the coordinates negative one, negative two; zero, negative one; and one, zero. The only one of these graphs which passes through these points is that given in option (b). And so that would be the answer. An alternative method of solving this question would be one involving the transformation of functions. We can consider the function ๐‘“ of ๐‘ฅ equals the cube root of ๐‘ฅ. And notice that the function that we were given is different because one has been subtracted from the output.

We can draw a quick sketch of the graph of ๐‘“ of ๐‘ฅ equals the cube root of ๐‘ฅ. We can then use the fact that if we transform a graph by subtracting ๐‘Ž units from the output for any real value of ๐‘Ž, then the graph will shift down by ๐‘Ž units. In this case, we were subtracting one from the output, so our graph of ๐‘“ of ๐‘ฅ equals the cube root of ๐‘ฅ will shift downwards by one unit. The graph which demonstrates this is the one given in option (b). And so that is the answer.

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