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Domain Restrictions and Functions Defined Piecewise - Problem 2
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
Let's graph another piecewise defined function. Here we have g(x) equals -3 for x between -4, and 0. 4x minus 3 for x between 0, and 2. 10 over x for x between 2, and 5.
Let's first get ourselves acquainted with the function by computing some values. G(-4). -4 falls into this interval right at the end point. So I'm going to use this definition for g(-4). It's -3.
G(0) falls into the center definition, because x equals 0 here. So it's 4 times 0 minus 3, also -3. G(2) well, 2 is the left end point of this interval, so g(2) is going to be 10 over 2 which is 5. G(5), 5 is the other end point of this interval. We'll get 10 over 5 which is 2.
Let's graph this function, starting with the -3 part which is for x between -4, and 0. Now notice that we're not going to include this right end point, x equals 0 just from -4 up to, but not including 0. So -3 is going to be about here. We'll start at -4, and we'll go to the right. So this is my point -4, -3. Then we have an open circle here.
The second part is this 4x minus 3 part. Well this has a y intercept of -3, and a slope of 4. We're going to go for two units, and stop just short of 2. So this piece actually starts here. So it fills in the hole, it's got a slope of 4, so it goes up 4 over 1. Up 4 over 1. So it stops at the point 2, 5. This point here. Actually that should be open 2, 5. I'll just draw the line connecting those two points. It looks like this.
Finally, the third piece. We've already discovered that g(2) is going to equal 5. So the third piece actually starts where the second piece left off. It's a 10 over x graph. This is like a vertically stretched version of the y equals 1 over x graph. So we're going to get this kind of hyperbola shape between x equal 2, and x equals 5. So a little piece of it. So it starts up here at 2, 5 , and it ends at 2, 3, 4, 5 , 2, because that's the point we plotted. It will look something like this. That's our graph. We just put 5, 2 here. That's our graph in the piecewise function.
Now, an interesting question to ask is what's the domain and range? So I'm going answer this over here. Now the domain you can tell when you're looking at a piecewise function what the domain is immediately, because it's the combined set of numbers that goes from -4 to 5. The interval from -45. That's your domain.
What's the range? Let's take a look at the graph. The range is the set of y values that we get out of this graph. It goes as low as -3, and as high as 5, and it includes 5. So the range is going to be from -3 to 5. That's our domain and range, and that's our function y equals g(x).
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