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Which of the following graphs represents a function in which the domain is equal to the range?
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In this question, we’re given the graphs of five different functions.
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And we need to determine in which of these functions is the domain equal to the range.
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To answer this question, let’s start by recalling what we mean by the domain and the range of a function.
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First, the domain of a function is the set of all input values for our function.
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Second, we recall the range of a function is the set of all output values for the function given its domain.
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We need to determine in which of these functions is the domain equal to its range.
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And since we’re given the graph of a function, let’s recall how we find the domain and the range of a function from its graph.
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We do this by remembering that the 𝑥-coordinate of any point on our graph tells us the input value of 𝑥.
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And the corresponding 𝑦-coordinate gives us the output value for that input value of 𝑥.
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And therefore, the domain of a function is the set of all 𝑥-coordinates of points on its graph.
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And the range of a function is the set of all 𝑦-coordinates of points which lie on its graph.
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So to answer this question, let’s find the domain and range of each of the five options.
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Let’s start with option (A), and we’ll start with its domain.
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First, we can see the lowest value of 𝑥 which is a valid input is when 𝑥 is equal to one because the point with coordinates one, zero is the point with the lowest 𝑥-coordinates which lies on this curve.
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And for any value greater the one, we can see there’s a point on the curve with this value as an 𝑥-coordinate.
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For example, the vertical line 𝑥 is equal to five intersects the curve.
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Therefore, the domain of this function is all values greater than or equal to one.
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And We can find the range of this function in the same way.
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We start by finding the lowest 𝑦-coordinate of a point which lies on the curve.
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In this case, this is zero.
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And in fact, we could just stop here because zero is not an element of the domain of our function.
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So the domain and range are not equal.
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So the answer is not option (A).
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However, we can also find an expression for the range of this function.
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We can see for any value greater than or equal to zero, there is a point on the curve where this is a 𝑦-coordinate.
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For example, the horizontal line 𝑦 is equal to six intersects our curve.
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Therefore, the range of this function is all values greater than or equal to zero.
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We can follow the same process for all of the other options.
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In option (B), we can see the lowest value of an 𝑥-coordinate which lies on our curve is zero.
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And all of the values of 𝑥 greater than or equal to zero lie in the domain of our function.
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So the domain of this function is all values greater than or equal to zero.
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We can only input nonnegative values of 𝑥.
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However, if we find the range of this function, we again have the same problem.
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Negative one is an output of the function because it’s a 𝑦-coordinate of a point which lies on the curve.
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And in particular, this function evaluated at zero is equal to negative one.
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And negative one is not an element of the domain.
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Therefore, the domain and range of this function are not equal.
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And if we wanted to, we could just find the range of this function from its diagram.
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It’s all values greater than or equal to negative one.
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We get a very similar story in option (C).
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Let’s start by finding the domain of this function.
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First, we notice that is an endpoint of this graph.
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And we can see that this is the point with the highest 𝑥-coordinate which lies on the graph.
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This therefore tells us the highest possible input value of the function.
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The highest input value is zero.
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And once again, we can see the graph continues infinitely off to the left, which means all values of 𝑥 less than or equal to zero are possible input values of the function.
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Therefore, the domain of this function is all values less than or equal to zero.
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Next, we need to check if the domain and range of this function are equal.
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And once again, we’re going to use the fact that the range of this function is the set of all 𝑦-coordinates of points which lie on its graph.
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The lowest 𝑦-coordinate of a point which lies on its graph is the point zero, negative one.
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The lowest output of this function is negative one.
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So the domain and range of this function cannot be equal.
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So option (C) is not correct.
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However, we can directly find the range of this function from its diagram.
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All values greater than or equal to negative one are possible outputs of the function.
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So the range is the set of values greater than or equal to negative one.
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We will apply the same process to option (D).
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We’ll start by finding the domain of this function.
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We can see the lowest 𝑥-coordinate of a point on the line is the point zero, one with 𝑥-coordinate zero.
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And since the graph of this function continues infinitely in this direction, we can see that any possible value greater than or equal to zero is a possible input value for our function.
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The domain of this function is all values greater than or equal to zero.
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We can then use this to determine the range of the function.
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We can see the lowest 𝑦-coordinate of a point on the curve is one which also means the lowest output of the function is one.
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However, the domain of this function includes zero.
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Therefore, zero is not an element of the range of this function.
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The domain and range are not equal.
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This is enough to exclude this option.
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However, we can also find the range of this function from the diagram.
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On the diagram, we can see as 𝑥 approaches infinity, our function also approaches infinity.
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And therefore, from the diagram, we can see any possible value of 𝑦 greater than or equal to zero is a 𝑦-coordinate on the curve.
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The range of the function is all values greater than or equal to one.
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Finally, let’s determine the domain and the range in option (E).
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First, the range is the set of all 𝑥-coordinates of points which lie on the curve.
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We can see the lowest 𝑥-coordinate of a point on the curve is the point zero, zero.
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And since the graph of our function continues infinitely to the right, there is a point on the curve for every 𝑥-coordinate greater than or equal to zero.
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Another way of thinking about this is every vertical line in the form 𝑥 is equal to 𝑐, where 𝑐 is greater than or equal to zero, intersects the curve.
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And the vertical lines to the left of the vertical axis don’t intersect the line.
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Therefore, the domain is all values greater than or equal to zero.
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And we can see something very similar for the range.
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Remember, the range of the function is the set of all 𝑦-coordinates of points which lie on its curve.
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And from the diagram, we can see the lowest 𝑦-coordinate of any point on the curve is the point with coordinate zero, zero.
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And from the diagram, we can see the end behavior of our graph.
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As 𝑥 is approaching infinity, 𝑦 is unbounded.
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𝑦 is approaching positive infinity.
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So we can see from the diagram any possible value of 𝑦 greater than or equal to zero is the 𝑦-coordinate of a point on the curve.
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Therefore, the range of this function is all values greater than or equal to zero.
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Therefore, we were able to show of the five given options only the graph in option (E) represents a function which has a domain equal to its range.