Math Is Fun Forum

jadewest

Member

Registered: 2021-02-20

Posts: 44

Range and domain of functions

Hello,

I can't solve this exercise.

What is the domain and range of the function:

y = x^2 / (x^2-16)

Thank you so much,
Jade

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,403

Re: Range and domain of functions

Hi,

The domain is The domain is x ∈ (-∞, -4) ∪ (-4,4) ∪ (4,+∞).

The range is y ∈ (-∞,1) ∪ (1,+∞).

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It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

jadewest

Member

Registered: 2021-02-20

Posts: 44

Re: Range and domain of functions

Hello,

I have completed these exercises about finding the domain and range. Are they correct?

3. y = 20 - 2x^2

Domain: (-∞, +∞)
Range: (-∞, 20)

4. y = 25(-7x-4)^64

Domain: (-∞, ∞)
Range: (0,  ∞)

Thank you so much,
Jade

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Range and domain of functions

Hi Jade

I think the key is to have a good understanding of the graph. I am unable to post a good picture a the moment but you can go to the  MIF function graphed to make your own.

But the following techniques are worth learning anyway.

Firstly what is y when x=0 and what is x when y=0?
Only x=y=0 works so the curve has only the origin as a point on either axis.

Note next only even powers of x. Say x=5. Then y= 25/9.
When x=-5 y is 25/9 again. This happens for all x if (p,q) is on the curve so is (-p,q)

So the y axis is a reflection line for the curve

Next look for asymptote. This means a line on the graph paper that the curve approaches but never touches.
There are 3 such lines. First at x=-4. When x is just right of -4, say -3.9999 y is large and negative. We say y tends to - infinity as x tends to -4 from the right.
When x is just left of -4, say -4.000001, y is large and positive. We say y tends to + infinity as x tends to -4 from the left.

More in next post

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Range and domain of functions

As this is long I thought I'd post in sections.

x=4 is another asymptote with y tends to infinity as x tends to +4 from the right and -infinity as x tends to 4 from the left.

Now what happens as x tends to infinity?
1000000^2/(1000000^2-16) is pretty close to 1 but just over.we say as x tends to infinity y tends to 1 from above. So y=1 is another asymptote. Same as x tends to -infinity.

Between -4 and +4 y is always negative except at (0,0). We know about the reflection symmetry so now we can put together as sketch of the curve.
Far left the curve rises up from the line y=1and curves uo towards the line x=-4. Similarly far right it rises from the line y=1 and curves left and up heads towards x=4.
Right of x=-4 it comes up from -infinity rises to touch the axis at x=0 then curves down towards -infinity again at x=+4.

Now to answer the question. The domain is just the values x can take: from -infinity to -4 not including -4, then all points to +4, again not including +4, then all points after that.

For the range anything negative up to and then including 0.  We can write this as (-infinity,0] the square bracket

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Phrzby Phil

Member

From: Richmond, VA

Registered: 2022-03-29

Posts: 50

Re: Range and domain of functions

I am an experienced math teacher and tutor.

New to this very interesting forum, I am rather astonished and disappointed that some knowledgeable responders just give a student the answer, as above, rather than ask the appropriate questions to the poster to guide the student to the answer.

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World Peace Thru Frisbee

Bob

Administrator

Registered: 2010-06-20

Posts: 10,621

Re: Range and domain of functions

hi Jade,

When you posted I was on holiday just using a phone and the hotel wifi.  I don't know how to save a screen shot on a phone (must learn that now) so that I could include a picture of the graph.  Now I'm home I can; so here it is. Hope it makes things clearer.

The function is shown in blue and the three asymptotes in red.  As you can see, the function curves towards these red lines but never actually reaches any of them (because you cannot plot a point that is 'at infinity').

Having a graph should make it easy to work out the domain and the range just by looking at which coordinates are possible and which are not.

To answer your later post more carefully, if an endpoint is part of the range or domain then the convention is to indicate this with a square bracket.  If the endpoint is not, then use a round bracket. The answers you give in your second post need modification as both 20 and zero are included endpoints.

to Phrzby Phil.  My apologies for not responding sooner.  The hotel bandwidth was inadequate for all the guests using it and I was finding it hard to accomplish anything much of the time.

I, too, am a teacher and I agree with you.  If a poster just asks for answers they don't get that from me.  If they seem genuine in wanting to understand the work and to do it themselves, then I might provide one answer as a model and then expect them to have a go themselves at the rest.  Sadly, that is often the last I hear from the poster.

In this case I know that Jade is a conscientious student who wants to properly understand each topic.  My problem was that an answer without explanation had already been posted. Using my admin powers I can delete a post or edit content, but I am very reluctant to do this.  I decided that it would be best to show Jade how to create a graph sketch so she would be well armed to tackle further questions like this one.  But I found it really hard to post properly due to the bandwidth issue.  I couldn't get the grapher to load at all and didn't know how I could grab a screen shot anyway.

After I had told her how to construct the graph sketch I realised there was another problem. 

The range is y ∈ (-∞,1) ∪ (1,+∞).

My worry then was that Jade might think that is correct, when, in fact, no y values between 0 and 1 are possible. Between x=-4 and +4 no positives are possible for y.  So I thought I'd better post my answer and also use the opportunity to say about () and [].

My ideal way forward now would be if Jade posted back saying whether she 'gets it' or not.  I would then be happy to give her another example as 'homework' to show she has mastered the topic.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob