A piecewise function is a function which is defined by multiple subfunctions, each subfunction applying to a certain interval of the main function's domain (a subdomain). Piecewise is actually a way of expressing the function, rather than a characteristic of the function itself, but with additional qualification, it can describe the nature of the function. For example, a piecewise polynomial functions: a function that is a polynomial on each of its subdomains, but possibly a different one on each.

Note that, in this case, we have three different “formulas” for

Note that, in this case, we have three different “formulas” for

*f*(*x*): -*x*+ 1, 2 and*x*4. But remember, with a function, when you evaluate f(x), you should get ONE ANSWER—not three. Otherwise, you don’t have a function.^{2}-
How do you know which answer you should have
(i.e., which formula you should use)? The “formula” is chosen using the
right-side of the piecewise-defined function (the “if …” part).

For example, suppose we wanted to find

*f*( − 3). Since − 3 < − 1, we will use the first “formula”:*f*( − 3) = - (-3) +1 = 4.

If we want to find

*f*(5), since 5 > 3, we use the third “formula”:*f*(5) = 5

^{2}- 4 = 21.

What about finding

*f*(1)? Note that in the interval (-1,3) the value of function is 2.
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