# Piecewise defined functions and integrals

• Nov 29th 2007, 05:06 AM
ebonyscythe
Piecewise defined functions and integrals
I'm not even sure where to begin on this one, or how f(x) and g(x) are related...

Let
------{0 if x< 0
------{x if $\displaystyle 0 \leq x \leq 1$
f(x) = {2-x if $\displaystyle 1 < x \leq 2$
------{0 if x> 2

(excuse the mess, but I've no idea how to do the piecewise functions with math tags.)

a) Find an expression for g(x) similar to the one for f(x), i.e., a piecewise defined function.

b)sketch the graphs of f(x) and g(x) on the same coordinate system.
• Nov 29th 2007, 05:20 AM
colby2152
g(x) is one function that looks like f(x)... it starts at (0,0) and goes to (1,1) and then down to (2,0) and stops. This is the nature of absolute value of a function, notably:

$\displaystyle g(x)=-|x-1|+1$ for the domain of x: (0,2)

HINT: start with $\displaystyle g(x) = |x|$ and then flip it, shift it horizontally, and then vertically
• Nov 29th 2007, 05:34 AM
ebonyscythe
I'm a little confused about the first part, how I know that g(x) is an absolute value function?
• Nov 29th 2007, 05:43 AM
colby2152
Quote:

Originally Posted by ebonyscythe
I'm a little confused about the first part, how I know that g(x) is an absolute value function?

Because of the shape of f(x), it basically looks like the karat symbol that is above your six on the keyboard.
• Nov 29th 2007, 06:02 AM
ebonyscythe
Well I understand that, it's just I'm not sure how the two functions are connected. How can I conclude that g(x) is an absolute value from f(x)? Is it because g(x) is the integral of f(x)?
• Nov 29th 2007, 06:12 AM
topsquark
I'm beginning to believe in psychic communication...

Where, in the original post or anywhere else, is g(x) defined??

-Dan
• Nov 29th 2007, 06:12 AM
CaptainBlack
Quote:

Originally Posted by ebonyscythe
I'm not even sure where to begin on this one, or how f(x) and g(x) are related...

Let
------{0 if x< 0
------{x if $\displaystyle 0 \leq x \leq 1$
f(x) = {2-x if $\displaystyle 1 < x \leq 2$
------{0 if x> 2

(excuse the mess, but I've no idea how to do the piecewise functions with math tags.)

a) Find an expression for g(x) similar to the one for f(x), i.e., a piecewise defined function.

b)sketch the graphs of f(x) and g(x) on the same coordinate system.

What do you know about g(x), I see no information on how it is related to

RonL
• Nov 29th 2007, 06:20 AM
ebonyscythe
Whoops, I did cut off the question... sorry about that.

$\displaystyle g(x) = \int_0^x f(t)dt$

That's probably pretty important, yeesh.
• Nov 29th 2007, 06:50 AM
ebonyscythe
Anyone? I gotta leave for Psych soon, so I won't be around much longer.
• Nov 29th 2007, 07:01 AM
Jhevon
Quote:

Originally Posted by ebonyscythe
Anyone? I gotta leave for Psych soon, so I won't be around much longer.

By the second fundamental theorem of calculus, if $\displaystyle g(x) = \int_0^x f(t)~dt$ it means that $\displaystyle g'(x) = f(x)$, that is, the function $\displaystyle f$ is the derivative of the function $\displaystyle g$. so to get $\displaystyle g$, integrate all the pieces of $\displaystyle f$