piecewise functions?

Question

i dont get them- are there any websites that show step by step?

Answer 1

"Piecewise functions" are functions comprised by other functions defined by certain intervals. Thus, you can have a number of functions representing one piecewise function.

Here is an example of a piecewise function

f(x) = sqrt(x); Interval = 8 > x
f(x) = x^2-log(x) Interval = 1 <= x <= 8
f(x) = 3; Interval = -5 <= x < 1
f(x) = sqrt(-x); Interval = x < -5

Basically, for each interval, you would graph the function defined by that interval.

Answer 2

I'm not aware of any, so perhaps I shouldn't be answering, but piecewise functions, while they can be confusing, really aren't hard.

Think of a function as being like a length of string. If the function is defined over all the reals, then the string is infinitely long. To make a piecewise function, what you're doing is taking pieces of different strings and tying them together at various points to make a single piece. To do this correctly, you'd need to know what strings to use and where to put them, and that's all you need to give for a piecewise function.

For instance:

f(x) = x^2
g(x) = 2x
h(x) = -2x

There we have three pieces of string. Let's take the left side of h, the right side of g, and the middle of f and tie them all together. So we get a piecewise function - call it p(x) - where

p(x) = -2x for x <= -2
= x^2 for -2 < x < 2
= 2x for x >= 2

This function will be a line with negative slope up to x = -2. From there, it's a parabola with vertex at the origin up until x = 2. After that, it's a line with positive slope. It's called a piecewise function because it's made up of pieces of other functions.

Does that make sense?

Answer 3

Piecewise functions are functions that are not continuous throughout the entire range of values. For example the piecewise function could be continuous for values ox between 0 and 5 and then be non continuous and then continuous again between 5+ and 10.

A good example of a piecewise continuous function is the step function or a function whose graph looks like a staircase. The function is continuous along each step, but then it jumps up to the next step and is thus discontinuous there. Then it is again continuous until it jumps up to the nest step, and so on.