# Piecewise Function - Domain & Range

• Sep 21st 2009, 08:09 AM
BeSweeet
Piecewise Function - Domain & Range
For the function
_____{x+3 if -5≤x<2
f(x)={x^2 if 2≤x≤4

I have to sketch the graph, determine the domain and range of f(x), and determine f(2). Help please? (Headbang)
• Sep 21st 2009, 08:22 AM
red_dog
$f(x)=\left\{\begin{array}{ll}x+3 & ,-5\leq x<2\\x^2 & ,2\leq x\leq 4\end{array}\right.$

The domain is $[-5,4]$.

To find the range we can use the following statement abou function:

If $f:A\to B$ and $X, \ Y$ are two subsets of A, then

$f(X\cup Y)=f(X)\cup f(Y)$

In this case let $X=[-5,2), \ Y=[2,4]$

We have $f(X)=[-2,5), \ f(Y)=[4,16]$.

Then the range is $[-2,5)\cup[4,16]=[-2,16]$
• Sep 21st 2009, 08:34 AM
BeSweeet
Quote:

Originally Posted by red_dog
$f(x)=\left\{\begin{array}{ll}x+3 & ,-5\leq x<2\\x^2 & ,2\leq x\leq 4\end{array}\right.$

The domain is $[-5,4]$.

To find the range we can use the following statement abou function:

If $f:A\to B$ and $X, \ Y$ are two subsets of A, then

$f(X\cup Y)=f(X)\cup f(Y)$

In this case let $X=[-5,2), \ Y=[2,4]$

We have $f(X)=[-2,5), \ f(Y)=[4,16]$.

Then the range is $[-2,5)\cup[4,16]=[-2,16]$

Wow... That is completely confusing. Never seen anything that looks anything like that stuff.
• Sep 21st 2009, 08:47 AM
red_dog
Quote:

Originally Posted by BeSweeet
Wow... That is completely confusing. Never seen anything that looks anything like that stuff.

Then let's do it in another way.

$-5\leq x<2$

Add 3 to all members: $-2\leq x+3<5$

Then, if $x\in[-5,2)$ then $f(x)\in[-2,5)$

$2\leq x\leq 4$

Square all members: $4\leq x^2\leq 16$

Then, if $x\in[2,4]$ then $f(x)\in[4,16]$.

So, the range is $[-2,5)\cup[4,16]=[-2,16]$

Is it better now?
• Sep 21st 2009, 09:08 AM
stapel
Quote:

Originally Posted by BeSweeet
For the function
_____{x+3 if -5≤x<2
f(x)={x^2 if 2≤x≤4

I have to sketch the graph...

Sketch y = x + 3, and then erase everything before x = -5 and after x = 2. Make sure to draw a filled-in circle for the left-hand endpoint and an "open" circle for the right-hand endpoint.

Then sketch y = x^2, and erase everything before x = 2 and after x = 4. Make sure to draw filled-in circles for each of the endpoints.

Quote:

Originally Posted by BeSweeet
...determine the domain and range of f(x)...

The domain is given: it's the x-values for which the function is defined.

To find the range, look at your graph. Which y-values are covered by this graph? (If you "collapsed the graph sideways onto the y-axis, which portions would be covered?)

Quote:

Originally Posted by BeSweeet
...and determine f(2).

The function is defined for x = 2. So look at the function rule, find the half which is defined for x = 2, and plug 2 in for x in that half's rule.

(Wink)
• Sep 21st 2009, 03:11 PM
BeSweeet
Quote:

Originally Posted by stapel
Sketch y = x + 3, and then erase everything before x = -5 and after x = 2. Make sure to draw a filled-in circle for the left-hand endpoint and an "open" circle for the right-hand endpoint.

Then sketch y = x^2, and erase everything before x = 2 and after x = 4. Make sure to draw filled-in circles for each of the endpoints.

The domain is given: it's the x-values for which the function is defined.

To find the range, look at your graph. Which y-values are covered by this graph? (If you "collapsed the graph sideways onto the y-axis, which portions would be covered?)

The function is defined for x = 2. So look at the function rule, find the half which is defined for x = 2, and plug 2 in for x in that half's rule.

(Wink)

I'm still unsure at how you are supposed to graph this thing. I don't understand the domain & range either. I do understand the f(2) thing. The answer for that part is 4, right?
• Sep 21st 2009, 03:35 PM
stapel
Quote:

Originally Posted by BeSweeet
I'm still unsure at how you are supposed to graph this thing.

To learn how to graph linear equations, such as y = x + 3, try here.

to learn how to graph quadratics, such as y = x^2, try here.

They were supposed to have covered this material way before moving on to piecewise functions, etc. (Wondering)
• Sep 21st 2009, 03:53 PM
BeSweeet
Quote:

Originally Posted by stapel
To learn how to graph linear equations, such as y = x + 3, try here.

to learn how to graph quadratics, such as y = x^2, try here.

They were supposed to have covered this material way before moving on to piecewise functions, etc. (Wondering)

I get it!
a. Figured out how to sketch it.
bA. For the domain, is it (-, -5)∪(-5, 2)∪(2, 4)∪(4, )... Something like that?
bB. For the range, I'm still not getting that part.
c. Determining $f(2)$ is simple. Should be 4, right?