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Consider the function π of π₯ equals the square root of π₯ minus one.
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Which of the following graphs could represent π of π₯?
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Using the graph of π of π₯, find its domain.
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Using the graph of π of π₯, find its range.
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In order to identify the correct graph of π of π₯ equals the square root of π₯ minus one, letβs remind ourselves what the function the square root of π₯ looks like when graphed.
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Itβs the inverse of the function π of π₯ equals π₯ squared, but we restrict it to make sure that itβs one to one.
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And so it looks a little something like this.
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In order to use this graph to identify the correct graph of the square root of π₯ minus one, letβs recall a function transformation.
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Suppose we have the graph of π¦ equals π of π₯.
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This is mapped onto the graph of π¦ equals π of π₯ minus π for some real constant π by a transformation by the vector π, zero.
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In other words, the graph is moved π units to the right.
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So in this case, we are subtracting one from the π₯ inside the square root symbol, meaning that the graph of π of π₯ must be translated one unit to the right to map onto π of π₯.
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So it will intersect the π₯-axis at one and look a little something like this.
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If we look carefully at all five of our graphs, we can observe that that is graph (B).
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The next part of this question asks us to use the graph to find the domain of our function.
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And so we recall that the domain is a set of possible inputs to the function.
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In other words, for some function π of π₯, what π₯-values can we substitute into that function to get real outputs?
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With this definition in mind, it follows that we can use the spread of π₯-values on our graph to find the domain of our function.
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Now, if we look at the spread of π₯-values on graph (B), we see that they start at π₯ equals one and extend to positive β.
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So π₯ can take values greater than or equal to one.
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In set notation, we say that the domain is the set of values in the left-closed, right-open interval from one to β.
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And so weβre ready for the final part of this question.
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It asks us to find the range by using the graph of π of π₯.
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The range of the function is the set of possible outputs to the function when the values from the domain are substituted in.
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In other words, given a function π¦ equals π of π₯, the range is the set of possible π¦-values.
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And so we can look at the spread of possible π¦-values in the vertical direction on our graph to establish the range.
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On graph (B), we see that the π¦-values start at zero and they extend towards β.
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Now, whilst it might look like they reach some sort of limit, we know that this is not true since the square root of β minus one is simply β.
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So the range, the set of possible outputs, is all values greater than or equal to zero.
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Using set notation, the range of the function is the left-closed, right-open interval from zero to β.