SOLUTION: Find the range with algebra (no graphing) of f(x)/g(x) if f(x) = sqrt{4 - x^2} g(x) = sqrt{9 - x^2}

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Question 1208157: Find the range with algebra (no graphing) of f(x)/g(x) if
f(x) = sqrt{4 - x^2}
g(x) = sqrt{9 - x^2}
Found 4 solutions by Edwin McCravy, mccravyedwin, greenestamps, math_tutor2020:
Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!


The minimum value is when the numerator is 0, or when , which
is when  = 0


                        4/9 
9 + 0x - x2) 4 + 0x -   9x2 
             4 + 0x - 4/9x2
                    -77/9x2




Thus the maximum value of  is when we subtract the least
amount possible from the 4/9 under the radical, which is when x=0:

 

Thus the range is  or in interval notation: [0,2/3]

Edwin
Answer by mccravyedwin(407)   (Show Source): You can put this solution on YOUR website!

I made corrections on the above.

Edwin
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!




Use the property (for A and B positive):



Rewrite the radicand:





This expression is the square root of something less than 1.

Its minimum value is 0, when the radicand is zero:






The lower end of the range of the given function is 0, when x is 2 or -2.

The maximum value of the radicand is when is a minimum -- i.e., when the smallest possible value is subtracted from 1. That is when x is 0; the upper end of the range of the given function is



ANSWER: the range of the given function is [0,2/3]
Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

The other tutors have done a great job.
I know the instructions specifically state "no graphing", but let's look at a graph anyway to help verify the range is [0, 2/3] aka



The green curve is
The blue horizontal line is y = 2/3 to help show the curve maxes out at this y value.

For some reason, the built-in algebra.com graphing tool shown above is glitching a bit.
The part of the curve on the right needs to extend down to the point (2,0).
Perhaps a better look would be to use something like Desmos
https://www.desmos.com/calculator/nxfzbzullv

GeoGebra is another useful graphing tool.
There are many others.
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