WEBVTT
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Determine the range of the function represented by the following graph.
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In this question, we’re given the graph of a function and we need to use this to determine the range of this function.
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To do this, let’s start by recalling what we mean by the range of a function.
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It’s the set of all possible output values of the function given its domain.
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And we need to determine this set by using the graph of this function.
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So let’s recall how the input and output values of a function relate to its graph.
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We can do this by remembering when we sketch the graph of a function, the 𝑥-coordinate of a point on the curve represents the input value and the corresponding 𝑦-coordinate represents the output value of the function for that 𝑥-input.
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In particular, this means the 𝑦-coordinate of any point on the curve is an output value of the function.
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Therefore, since the range of the function is the set of all possible output values of the function, it’s also the set of all 𝑦-coordinates of points which lie on the curve.
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This means we can determine the range of this function by just finding the set of all 𝑦-coordinates of points which lie on the curve.
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We can do this from the graph.
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Let’s start by finding the lowest possible 𝑦-coordinate of a point on the curve.
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From the diagram, we might be tempted to think that this is negative three.
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However, this is a hollow dot.
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So remember, this means the function is not defined at this point.
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So the function never outputs negative three.
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However, we can see from the graph of this function we can get closer and closer to negative three.
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Therefore, the function can output any value close to negative three from above but not equal to negative three.
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And since this is the lowest output value of the function, we can also say that the function does not output any value less than or equal to negative three.
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Let’s now consider output values greater than negative three.
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And to do this, we need to recall when there are arrows on our function, this means it continues infinitely in this manner.
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So because there are arrows on both sides of this function, we know both of these straight lines continue indefinitely.
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We can then use this to determine all values greater than or equal to negative three are possible output values of the function.
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We can see this by noting any possible value greater than or equal to negative three is a 𝑦-coordinate of a point which lies on the curve.
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For example, one is a 𝑦-coordinate of a point which lies on the curve.
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The only point we might be worried about is two because there’s a hollow dot on our curve at two.
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However, we can also see that there is another point on the curve with 𝑦-coordinate two.
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So two is a 𝑦-coordinate of a point which lies on the curve, which also means it’s an output value of the function.
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Finally, since our lines continue indefinitely in both directions, we can also see that these continue up to ∞.
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So this process will hold for all values bigger than negative three.
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Therefore, the output values of this function are all values greater than negative three.
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We need to write this as a set, and this is the open interval from negative three to ∞.