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Determine the domain and the range of the function π of π₯ is equal to six when π₯ is less than zero and π of π₯ is equal to negative four when π₯ is greater than zero.
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In this question, weβre given a piecewise-defined function π of π₯ and the graph π¦ is equal to π of π₯.
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We need to use this to determine the domain of π of π₯ and the range of π of π₯.
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So, letβs start by recalling what we mean by the domain and range of a function.
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First, the domain of a function is the set of all input values for that function.
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Second, the range of a function is the set of all output values for our function given its domain.
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Letβs start by finding the domain of our function π of π₯.
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And thereβs two different ways of doing this.
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First, we can just look at the piecewise-defined function π of π₯.
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We want to find all of the possible input values of our function.
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And we can see from the piecewise definition of our function, our function outputs six whenever π₯ is less than zero and our function outputs negative four when π₯ is greater than zero.
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These two inequalities are called the subdomains of our piecewise function.
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They tell us the input values of π₯ for which our function corresponds to the given subfunction.
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We can see weβre allowed to input any value of π₯ less than zero or any value of π₯ greater than zero.
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This is any value of π₯ which is not equal to zero.
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So, we can input any value of π₯ which is not equal to zero into our function.
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And remember, we write the domain as a set.
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So, the domain of this function is the set of all real numbers minus the set including zero.
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However, this is not the only way we can find the input values for our function since weβre given a graph of the function.
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Remember, in a graph, the π₯-coordinate of any point on our curve tells us the input value of π₯ and the π¦-coordinate tells us the corresponding output of the function.
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For example, when π₯ is equal to five, we can see the point with coordinates five, negative four lies on our graph.
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Therefore, we can input the value of five into our function.
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And π evaluated at five is negative four.
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So, five is in the domain of our function.
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Another way of saying this is thereβs a point of intersection between the vertical line π₯ is equal to five and the given graph.
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This is sometimes called the vertical line test.
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We consider vertical lines and look for points of intersection.
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If thereβs no points of intersection, then that value does not lie in the domain of our function.
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We can consider sliding a vertical line across our diagram.
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For example, if π₯ is equal to 1.5, we can see thereβs a point of intersection with our graph.
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So, π₯ is equal to 1.5 is in the domain of our function.
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We might be worried about values of π₯ greater than nine since it appears that this does not intersect our function.
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However, we can notice the end of our graph has an arrow, and this notation means that our graph continues infinitely in this direction.
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The same will be true of the other arrow on our diagram.
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So, any vertical line on the positive part of our π₯-axis will intersect the graph.
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In fact, the same is true on the negative part of our π₯-axis.
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Any vertical line will intersect the graph.
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However, weβve not considered what will happen when our value of π₯ is equal to zero.
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We can see in our diagram there are two points which appear to be on our graph when π₯ is equal to zero.
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However, both of these are hollow circles, and these mean that our function is not defined at this point.
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So, the line π₯ is equal to zero does not intersect our graph, so zero is not in the domain of our function.
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Therefore, weβve shown graphically the domain of our function is the set of all real values excluding zero.
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Letβs clear some space and then determine the range of our function.
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Remember, thatβs the set of all possible output values of our function given the domain.
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And once again, thereβs two different ways of doing this.
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We can do this directly from the piecewise definition of π of π₯.
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We can see when our input value of π₯ is less than zero, our output is a constant value of six.
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And when π₯ is greater than zero, our output value is a constant value of negative four.
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So, in fact, there are only two possible output values of our function.
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And the range is the set of these possible output values.
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The range of our function is the set containing negative four and six.
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We could stop here; however, it is also important to be able to determine the range of a function from its graph.
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This time, remember, the output values of our function are represented by their π¦-coordinates on the graph.
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So, we can determine the range of a function by determining all possible π¦-coordinates of points on the curve.
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However, we can see in our diagram, there are only two possible π¦-coordinates of any point on the curve.
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Either it has a π¦-coordinate of six or it has a π¦-coordinate of negative four, confirming that the range is the set containing negative four and six.
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Therefore, we were able to show for the function π of π₯ is equal to six when π₯ is less than zero and π of π₯ is equal to negative four when π₯ is greater than zero, the domain of this function is the set of real numbers excluding zero and the range of this function is the set containing negative four and six.