Piecewise Defined Functions
Piecewise defined functions are functions which symbolically define two or more formulas. For Example an absolute value graph is an example of a piecewise function.
The Function above means that any number less than zero is plugged into the top function and any number greater than or equal to zero is plugged into the bottom function.
Examples:
When these functions are graphed together the look like this:
If you have a graphing calculator you can go to the "Y=" button and type in:
After this press the graph button and you can see your graph. If you need to find a certain point on the graph you click "second" then "graph" and you find the table of points.
The Difference Quotient
The Difference Quotient is the slope of a line through the points (x, f(x)) and ( x + h, f(x + h) ).
The Difference Formula is:
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h
An example is:
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h
Distribute through the problem
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h
Cross out the terms that equal zero
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h
Factor out an h
Extra Notes:
- There should always be terms that cancel out after you distribute through the problem
- You should not, in most cases, get an answer of 1
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