# Thread: Piecewise defined functions and integrals

1. ## 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.

2. 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

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

4. 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.

5. 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)?

6. I'm beginning to believe in psychic communication...

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

-Dan

7. 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

8. 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.

9. Anyone? I gotta leave for Psych soon, so I won't be around much longer.

10. 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 $\displaystyle f$