Question 78018: Graph the function and state the domain and range.
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website!
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The easiest way to graph this is to plot some values. If you think about it, one value to
look at is when x = +2. Why? Because when x = 2 the term inside the absolute value signs
becomes +2 - 2 or zero, and therefore, y = 0.
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Next, you can find out what the value of y is when the graph crosses the y-axis. You can do
that by letting x = 0. (Think about it ... by setting x = 0, the corresponding value of
y has to be on the y-axis.) When x = 0 the quantity inside the absolute value signs becomes
0 - 2 = -2. And when you take the absolute value of -2 you get +2. So when x = 0 the corresponding
value of y is +2.
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Next try a few negative values of x. Let's let x = -5 and you get:
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Now let x = -8 and you get:
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Next let's look at a few positive values of x. Let's give x a value of +5. The equation
becomes:
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Now let's give x a value of +8. The equation then becomes:
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In summary, we know that the following are points on the graph:
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(+2, 0); (0, +2); (-5, +7); (-8, + 10); (+5, +3), (+8, +6)
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You can get more points if you like by doing the same sort of process with different
values of x. Let's draw the graph and then discuss the domain and range. The graph is:
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By looking at the graph, you can see that there is no limit on the value of x. You can
assign to x any value from minus infinity to plus infinity. However, look at the values
that y can take (the range). You can get y = 0 and y = any positive value, depending
on what value you assign to x. But you can never get a value of y that is negative.
So you can say that y must be equal to or greater than zero. In summary the domain is:
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-oo < x < + oo (where oo stands for infinity)
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and the range is:
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0 <= y (where <= represents greater than or equal to)
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Hope this helps you to understand the concept of absolute value a little more. When you
take the absolute value of a quantity, the result is a positive value.
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