Problem 1
Domain = {3, -3, 7, -8}
Range = {7, -2, -5}
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The domain is the set of possible x inputs, while the range is the set of possible y outputs. Toss out any duplicates.
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Problem 2
Domain in words = set of all real numbers
Domain in set builder notation = [tex]\{x|x \in \mathbb{R}\}[/tex]
Domain in interval notation = [tex](-\infty, \infty)[/tex]
Range in words = set of real numbers equal to -4 or greater
Range in set builder notation = [tex]\{y|y \in \mathbb{R}, \ y \ge -4\}[/tex]
Range in interval notation = [tex][-4, \infty)[/tex]
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I recommend graphing out y = 3x^2-4 to see a parabola in which we can plug in any x value without restriction. The lowest point on this graph is (0,-4) which tells us the smallest value in the range is -4. There is no largest value in the range.
The square bracket for the interval notation range is to indicate to include -4 as part of the interval. In contrast, a curved parenthesis means to exclude the endpoint.
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Problem 3
Domain in words = set of nonzero real numbers
Domain in set builder notation = [tex]\{x|x \in \mathbb{R}, \ x \ne 0\}[/tex]
Domain in interval notation = [tex](-\infty, 0) \cup (0, \infty)[/tex]
Range in words = set of nonzero real numbers
Range in set builder notation = [tex]\{y|y \in \mathbb{R}, \ y \ne 0\}[/tex]
Range in interval notation = [tex](-\infty, 0) \cup (0, \infty)[/tex]
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The domain and range are the same in this case. Basically x can be anything but zero since you cannot divide by zero. We can show that y = 0 is not possible if we were to find the inverse of y = 2/x getting x = 2/y, so we see that y = 0 would lead to another division by zero error. These domain and range restrictions lead to the vertical and horizontal asymptotes.
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Problem 4
Domain in words = set of real numbers equal to 3 or greater
Domain in set builder notation = [tex]\{x|x \in \mathbb{R}, \ x \ge 3\}[/tex]
Domain in interval notation = [tex][3, \infty)[/tex]
Range in words = set of all real numbers
Range in set builder notation = [tex]\{y | y \in \mathbb{R}\}[/tex]
Range in interval notation = [tex](-\infty, \infty)[/tex]
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The graph shows the left most point being (3,0). So x = 3 is the smallest x value allowed. There is no max x value assuming the graph continues on forever to the right.
Meanwhile, the range is any possible y value because the graph goes on forever up and down.
