Find the domain and the range of ##f-3g##

  • #1
chwala
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Homework Statement
The real functions ##f## and ##g## are given by

##f(x)=x-3## and ##g(x)=\sqrt {x}##

Find the domain and the range of ##f-3g##
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Functions
Am refreshing on this,

For the domain my approach is as follows,

##(f-3g)x = f(x)-3g(x)##
##=x-3-3\sqrt{x}##.

The domain of ##f-3g## is given by ##f∩g = [{x: x ≥0}]##

We have

##y= x-3-3\sqrt{x}=(\sqrt x-\frac{3}{2})^2-\dfrac{21}{4}##.

The least value is given by; ##\left(\sqrt x-\dfrac{3}{2}\right)^2 =0##. This occurs when ##x=2.25##.

The range of ##f-3g## is the set ##[{y: y≥-5.25}]##

Your insight or correction is welcome.
 
Last edited:
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  • #2
Looks good. Maybe a bit complicated but good.
 
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  • #3
Maybe one observation that may help is that in the final formula, the x term will dominate in going to infinity, so that ##f-3g## will be unbounded. The domain can be determined somewhat simply as the intersection of the domains, while I doubt there's a reasonable conclusion for such formulas as linear combinations of functions.
 
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  • #4
I would advise against using notation like ##f\cap g##. Intersection is a set operation. ##f## can be regarded as a set of ordered pairs and saying "domain is ##f\cap g##" is confusing the reader.

Whenever both ##f## and ##g## are used to compute a new quantity, it automatically follows that both ##f## and ##g## are well defined, so the domain of interest must be intersection of the individual domains.
 
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