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Piecewise Functions Review

Ah, Calculus teachers love using piecewise functions!  They stay home Saturday nights dreaming up new ones...  just for you!

If you've loathed them before, no fear.  I've got a simple way to keep all the pieces straight.

Let's just dive in and do one:

Graph y = ( x + 2 ; x < 1 ... ( x - 1 )^2 ; x is greater than or equal to 1 ... It's in two pieces!

Each piece must live ONLY in its own neighborhood.

Let's put up a fence, so we don't make any mistakes:
 

the fence is vertical at x = 1 ... 1 : y = x + 2 lives on the left side of the fence ... 2 : ( x - 1 )^2 lives on the right side of the fence
 

Now, we just need to figure out who the fence owner is...

y = ( x + 2 ; x < 1 ... ( x - 1 )^2 ; x is greater than or equal to 1 ) ... This second guy has the "=", so he gets to live ON the fence

Let's graph part1 ):

y = x + 2 ... He does not get to sit on the fence... and he can't be in the neighborhood on the right side of the fence