# Thread: Piecewise Function - Domain & Range

1. ## 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?

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

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 $[-2,5)\cup[4,16]=[-2,16]$

3. 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.

4. 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?

5. 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.

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?)

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.

6. 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.

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?

7. 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.

8. 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.
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?