the folowing piecewise function f(x) ={x+4 , -1 , 2x they were listed under one another see next box for detailsfor x<-3 , for -3<x<5(there was a line under the < sign after the -3) for...

the folowing piecewise function f(x) ={x+4 , -1 , 2x

they were listed under one another

see next box for details

for x<-3 , for -3<x<5(there was a line under the < sign after the -3) for x >5

a. find the domain

b. range

c. intergers

d. if f continuous on its domain? if not state wher f is discontinuous

e. graph the function

Asked on by stargsd5

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mlehuzzah's profile pic

mlehuzzah | Student, Graduate | (Level 1) Associate Educator

Posted on

The intercepts are where either x or y (or both) are 0.

So the intercepts here are (0,-1) and (-4,0)

mlehuzzah's profile pic

mlehuzzah | Student, Graduate | (Level 1) Associate Educator

Posted on

 

a) The domain is the x's that are "allowed" to be plugged into the function. Some examples of when you can't plug x into a function are:

`1/(x-2)` (you can't plug in `x=2` because then you would be dividing by 0, which is undefined)

`"ln" x` (you can only plug positive numbers into ln, so this function is undefined at `x <= 0`

Things like that.

Your function doesn't have any of those problems. You can certainly take any number and add 4, or multiply by 2. Your function might, however, have a different problem. The way you've written it, you have:

`x+4` for `x<-3`
`-1` for `-3 <= x < 5`
`2x` for `x>5`

If I say I want to plug in `x=-4`, then:
`-4<-3` so we use the "less than -3 rule" which is `x+4`

If I say I want to plug in `x=-3`, then:
`-3 <= -3 < 5` so we use the "between -3 and 5, including -3 rule" which is `-1`

If I say I want to plug in `x=6`, then:
`6>5` so we use the "greater than 5 rule" which is `2x`

But what if I say I want to plug in `x=5`
As you've got it written, there is no "rule" for `x=5`

So, the domain here is "all real numbers except 5"

 


b) The range is all the possible y values (or `f(x)` values)

If `x<-3` then `y=x+4` So:

`y=x+4 < -3+4 = 1`

Thus, we can achieve any value of y less than 1, so all of these numbers are in the range.

If `-3 <= x <5` then `y=-1`, so -1 is also in the range. (but this actually doesn't tell us anything new, because -1 was already in the range: it could be obtained by plugging in -5)

If `x>5` then `y=2x>2*5=10` so all the numbers greater than 10 are in the range.

The range is:

`(-oo,1) uu (10, oo)`


c) Integers

I'm not sure what you mean by this. Possibly plug in some integer values of x and see what the corresponding y values are, to help make the graph?

Here is a table:

`[[x,f(x)],[-5,-1],[-4,0],[-3,1],[-2,-1],[-1,-1],[0,-1],[1,-1],[2,-1],[3,-1],[4,-1],[5,"undefined"],[6,12],[7,14]]`


d) It is not continuous on its domain. To see this, what happens as x gets close to -3? If `x<-3` and x is getting close to -3, then we use the "x+4" rule, and `f(x)` is getting close to 1. But if `x>-3` and x is getting close to -3, then we use the "-1" rule, and `f(x)` is getting close to (actually equals) -1. The two are not the same, so there is a "jump" or discontinuity at x=-3.

At x=5 there is also a discontinuity, because the function is undefined there. This does not count as a discontinuity on the domain, however, since 5 isn't in the domain.


e)

 

 

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stargsd5 | Student, College Freshman | (Level 1) Valedictorian

Posted on

thank you for the help I was suppose to also find the intercept for part 1c I am unsure on this can you help?

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