What is the Domain and Range of a Function with Given Constraints?

[preet]

I'm a little confused here. I have a function: f(x) = x^2 + 3
If the domain of the function is

-3 <= x <= 3

I need to find the range...

I think that the answer is 3 <= y <= 12... that can be found out by drawing the graph. But I want to find the range algebraically. It says in my textbook to turn the 'x' in the middle of the domain shown above into the respective f(x) equation (x^2)+3... and performing whatever action on x on the other sides of the inequality. This works in other functions, but I think I'm doing something wrong with this one:

1.-3 <= x <= 3
2.9 <= x^2 <= 9
3.12 <= x^2 + 3 <= 12

I think that because in step 1 I multiplied one value by (-3) I need to flip the greater than/equal to sign(s), but not sure how to do it... as you can see, by step 3, the wrong answer is reached. Can anyone show me my mistake? Thanks!

 

[Math Is Hard]

Hi preet - just curious...
how did you come up with that domain? what's the problem with x = -4 or x = 4?x = -5 or x = 5? etc?

 

[vsage]

The method you used only works if f(x) has a true inverse function which in this case it doesn't. This is because more than one x value exists for every y which is untrue of truly invertible functions. The function f(x) where x is lies on [0, $$\infty$$) does have a true inverse. Take the domain to be 0<=x<=3 and then you can apply the rules you used.

 

[Math Is Hard]

ah, nevermind.. preet must be referring to a restricted domain.

 

[shmoe]

To go from an inequality like a<b to a^2<b^2 you need the additional assumption that a and b are non-negative 0<=a<b. Otherwise it may not be true, -4<3 but (-4)^2>3^2.

You can go a couple ways, do 2 cases, one where x<=0 and one where x>0. Or you can recognize that -3<=x<=3 is equivalent to |x|<=3, and work with 0<=|x|<=3.

 

[preet]

The domain was given... anyways... I understand now. Thanks!

 

[preet]

I have another question related to functions... I thought I'd post it in here instead of starting a new thread...

A graph of y=g(x) is as follows:
(-4, 3), (-1, -2), (2, -2), (3, 1), (6, 1)

the graph doesn't show a relation... it just shows a function... and the graph isn't a plot, so it isn't discrete.

f(x) = -g (1/2x - 3) + 1

Anyways, the questions are:
a) Draw an input/output diagram for f.
b) Determine the domain and range of f.
c) Graph 'f'

All I can do so far is
a) [ add 6 ] + [ multiply by 2 ] + [ g ] + [ multiply by -1 ] + [ add 2 ]
b) I don't know how to figure this out... the fact that function g is in function f is really confusing me...
c) I don't know how to do this either... I was thinking that I have to apply the transformations in the input output diagram, but I want to figure out part b first.

TiA.