Question 1

Can there be a video for when y isn't a static number, when we have equations for x and y? Because I do not know how to get the range for those, nonetheless graph it.
 The only way I can think of doing it, is guess and check from the domain, but there has to be an easier way, is there?

Accepted Answer

This answer is a year late, I'm also new to this topic, but I'll try to answer. I don't know if you had done this or not by saying "guess and check for the domain". But anyway, for example, we had a piecewise function like this :
```
          / 2x+3 , 0 < x <= 2
f(x) = {  x+2 , 2 < x <= 6
          \ 3x , 6 < x <= 11
```
Find the range of each clause separately.

For ```
2x+3 , 0 < x <= 2
```, since this is a straight line (with a slope), you just need to find the y values for the endpoints.
If x were 0, ```
2*0+3 = 3
```
If x were 2, ```
2*2+3 = 7
```
What is the range of this clause? that would be ```
3 < y <= 7
``` or it can be expressed more 'mathy' as ```
y or the range of the clause = (3, 7]
``` (this means all real values from 3 to 7 (3 is not included, 7 included))

Now you can find the other clauses' y values the same way. You'll get the range of the clauses :

range of 2x+3 = (3, 7]
range of x+2 = (4, 8]
range of 3x = (18, 66]

I'm not sure if this is legit, but the range of the whole function is the union of sets (3, 7], (4, 8], (18, 66] or (3, 7] U (4, 8] U (18, 66]

Am I making myself clear? Please correct me if I made any mistakes. Hope this helps

Question 2

This answer is a year late, I'm also new to this topic, but I'll try to answer. I don't know if you had done this or not by saying "guess and check for the domain". But anyway, for example, we had a piecewise function like this :
          / 2x+3 , 0 < x <= 2
f(x) = {  x+2 , 2 < x <= 6
          \ 3x , 6 < x <= 11

Accepted Answer

If you want the person who submitted the question to actually see your response, you must use the "reply" link underneath the question.  This ties your response to the question and the person who posted the question gets notified.  If you don't do this, the chance of that person seeing your response is pretty low.

Question 3

If the lowest number is 0, how can -7 be an input value?

Accepted Answer

-7 is not an input value.  The function specifies f(x) = -7 if x is in the interval:  6 <= x <= 11.  This means -7 is an output value.

The function is undefined for x = -7.
Hope this helps.

Question 4

Can there be a problem like:
f(x)= 1 , 0<x≤2
        5 , 4≤x<6
        -7 , 6≤x≤11
Where here there is no function defined for 3, if there is how do we find the domain?

Thanks

Accepted Answer

The domain is a union of 2 sets of values.  In interval notation it would be written as:  
(0, 2] U [4, 11]

The range would be written as a set of 3 values = {-7, 1, 5}

Hope this helps.

Question 5

So -7 is the x and 6 is the y and then we will switch y's with 11, and make a line, correct?

Accepted Answer

No, inside the parenthesis, the left hand side is the y and the right hand side is the x.
When x is between (and including) 6 and 11, y is always equal to -7.
You can imagine that someone has collected data of temperature in one location every day for a month. The data shows that between day 6 and day 11, the temperature was -7 every day.
Hope that helps!

Question 6

I'm confused here: I remember learning that a function is such only insofar as there's one (and one) only output for a given input. Isn't this case different?

Accepted Answer

No. Each input creates only one output, so it is a function. Notice the domain restrictions for each step happen to use the same numbers, but the end/start numbers are included in one step and excluded from the other step. Thus, there is no case where one input creates 2 outputs.

Hope this helps.

Hope this helps.

Question 7

If the intervals for x were 0<x<2 and 4<x<6 (as integers) then would the domain be {1,6}?

Accepted Answer

There are a couple of issues with your result:

1) {1, 6} is roster notation for a set.  It tells you the set includes 2 numbers, the 1 and the 6.  Interval notation uses parentheses or square brackets.

2) Interval notation assumes the interval is across all real numbers within the interval.  It would not be limited to integers.  And, your two inequalities have a gap.  All the numbers from 2 up thru 4 are not included.

The best way to write your domain would be to use set builder notation: {x | x ε Integers, 0<x<2 and 4<x<6}

Question 8

Nice Video, I just don't know how to solve for range in questions like
     [1-x   x<1
f(x)=[
     [√x-1  x>=1

Accepted Answer

If I understand correctly, you are looking for the range for the following definition:
`f(x) = { 1-x    for x < 1`
`       { √(x-1) for x ≥ 1`

Here is what I would do:
`f(x) = 1-x ` generates values from -∞ to almost 0 (for x almost equal 1)
`f(x) = √(x-1) ` generates values from 0 (for x=1) to ∞.

The range is -∞ < x < +∞

Question 9

shouldn't it be (x,y) but it looks as if the y is i front. could someone please explain this to me?

Accepted Answer

If you are plotting a point the the format is (X ,y) yes, but when you write out a piecewise function, the rule for calculating the output (y) is on the left, and  the specific domain for that rule is on the right. For example, say that over the interval -5<x≤5, the rule is 2x+1, it is written like this: f(x){2x+1, -5<x≤5} and then you just take a random working value from the domain on the right, plug it in to the formula on the left, and you get your new output.:f(-5)= 2(-5)+1 = -9. since the rule description is not the point itself, it is not written like a point. Hope this helps!:)

Question 10

So would it be called a "piecewise function" or "piecewise defined function"

Accepted Answer

Generally they are called "piecewise functions".  I've never seen them referred do as "piecewise defined function", though the words imply the same thing.

Course: Functions 237+ > Unit 5

Worked example: domain & range of step function

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