You can't put a television in a blender and you wouldn't expect an elephant to come out of a gasoline pump.In math terms, a television isn't an allowable input for a blender; it's not part of a blender's domain. And an elephant isn't a possible output of a gasoline pump; it's not part of a gas pump's range.
Similarly, functions have certain numbers that are and aren't allowed as inputs, and other numbers that are and aren't possible as outputs. In this activity, you'll explore these notions using both dynagraphs and Cartesian graphs.
Open the GSP file.
1. On page one, select the output pointer and choose trace triangle from the display menu. Drag the input pointer back and forth.
Note: The output pointer leaves a trace of where it's been. Since the traces point to all the integers and never to any non-integer value, we say that "the range of f is all integers."
2. Go to page two of the GSP file.
Use the same technique from above to find the ranges of the four functions modeled.
When using technology, it's very important to know the limitations of that technology. You will now see that the method used above can be misleading in certain situations. You will then learn a more reliable method.
3. Go to page three of the GSP file.
Turn tracing on for the output pointer, then drag the input pointer, as in step two. For greater control, use the right and left arrow keys on you keyboard to drag one pixel at a time. Select the input pointer and choose Animate Pentagon from the Display menu. Repeatedly press the Decrease Speed button (the down arrow) on the Motion Controller until it's clear that the range of this function really is all real numbers. You may also want to go back to pages one and two of the sketch and convince yourself that your answers there were correct.
4. Go to page four of the GSP file.
Drag v's input pointer back and forth. What is v's domain? What is v's range? Why, based on its equation, are some numbers not part of v's domain?
The domain of w is all real numbers except for one particular value. the range of w is also all real numbers except for one particular value (a different value). What is the one value not in the domain of w? What is the one value not in the range of w?