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

[chwala]

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 $f-3g$

Relevant Equations

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.

 

[fresh_42]

Looks good. Maybe a bit complicated but good.

 

[WWGD]

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.

 

[nuuskur]

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.