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Old 21st December 2011, 01:30 PM   #1 (permalink)
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Range of composite functions.

hi all!

i have a problem understanding how to find the range for composite functions. for example,

QUESTION:
http://i1088.photobucket.com/albums/.../IMG_18591.jpg

ANSWER:
http://i1088.photobucket.com/albums/.../IMG_18601.jpg

my notes says how to find composite function range:
use the range of the first function as the new "domain" of the second function. with the new "domain," find the range under the second function.

it doesn't work for this question and also all the other questions i've done!

please help THANKS! .

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Old 21st December 2011, 02:04 PM   #2 (permalink)
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Re: Range of composite functions.

For the solutions written in the second image, the range of fg is done correctly.

Range of g =[1,∞ )

Domain of f =(0, ∞ )

Range of g intersecting Domain of f = [1,∞ )

This interval [1,∞ ) becomes the new domain for the graph of f, ie
the range corresponding to the domain of [1,∞ ) for the function f = [2,∞ )

Therefore, [2,∞ ) is the range of fg.

This approach definitely works for any question.

Hope this helps. Peace.


Last edited by whitecorp; 21st December 2011 at 02:05 PM.
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Old 21st December 2011, 04:18 PM   #3 (permalink)
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Re: Range of composite functions.

OK thanks! i see that it works for that question but it doesn't seem to work again for the question below.

QUESTION: http://i1088.photobucket.com/albums/...4/IMG_1856.jpg
GRAPH: http://i1088.photobucket.com/albums/...4/IMG_1861.jpg (quadratic graph is function f the other one is function g).
ANSWER: http://i1088.photobucket.com/albums/...4/IMG_1855.jpg

to find fg:
range of first function f is [-4,infinity)
range of f becomes domain of g
new range of g/range of fg = ???????

the answer is [-4,infinity).

howwww?????
i don't get it.

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Old 21st December 2011, 04:38 PM   #4 (permalink)
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Re: Range of composite functions.

You got it all in reverse.

Range of g = (0, ∞ )

Range of g becomes Domain of f

ie new range of f aka range of fg = [ -4, ∞ ) [You can look at the quadratic curve of f and recognise that we only consider all positive real values of x, hence the lowest y-value in this spectrum is -4 which coincides with the minimum value, while the highest y value stretches clearly to infinity.]

If you were finding the range of gf, then what you have proposed would be absolutely correct.

Hope this helps. Peace.


Last edited by whitecorp; 21st December 2011 at 04:39 PM.
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Old 21st December 2011, 04:43 PM   #5 (permalink)
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Re: Range of composite functions.

my notes said: use the range of the first function as the new "domain" of the second function. with the new "domain," find the range under the second function.

the first and second function thing is confusing me. for example if they want range of fg like just now, shouldn't i find the range of function f as my first step? (because my notes said that). however your workings showed that i should find the range of g as my first step.

or do they mean first and second as in how it is listed in the question?

sorry for the trouble


Last edited by blacktoast94; 21st December 2011 at 04:45 PM.
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Old 21st December 2011, 04:52 PM   #6 (permalink)
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Re: Range of composite functions.

Its ok no worries.

Consider fg
1st function is the one on the right which is g.
2nd function is the one on the left which is f.

To find range of fg, range of g becomes new domain of f. Just focus on what you have gotten for range of g in the first place which is an interval set like (a,b) . Now use this (a,b) as the new domain for the graph of f, and find the corresponding new range. This new range is the range of fg.


Consider gf
1st function is the one on the right which is f.
2nd function is the one on the left which is g.

To find range of gf, range of f becomes new domain of g. Just focus on what you have gotten for range of f in the first place which is an interval set like (c,d) . Now use this (c,d) as the new domain for the graph of g, and find the corresponding new range. This new range is the range of gf.

Hope this helps. Peace.

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