Question 128482
{{{y=4+sqrt(x-2)}}} Start with the given expression


Remember you cannot take the square root of a negative value. So that means the argument {{{x-2}}} must be greater than or equal to zero (i.e. the argument <font size=4><b>must</b></font> be positive)


{{{x-2>=0}}} Set the inner expression greater than or equal to zero


{{{x>=0+2}}}Add 2 to both sides



{{{x>=2}}} Combine like terms on the right side



So that means x must be greater than or equal to {{{2}}} in order for x to be in the domain


So the domain in set-builder notation is

*[Tex \LARGE \textrm{\left{x|x\ge2\right}}]


So here is the domain in interval notation: [2,*[Tex \LARGE \infty])





Now the endpoint {{{x=2}}} of the domain corresponds to the endpoint of the range. So simply plug in {{{x=2}}} into {{{y=sqrt(x-2)+4}}}



{{{y=sqrt(x-2)+4}}} Start with the given function



{{{y=sqrt(2-2)+4}}} Plug in {{{x=2}}}



{{{y=sqrt(0)+4}}} Subtract



{{{y=0+4}}} Take the square root of zero to get zero



{{{y=4}}} Add



So when {{{x=2}}}, {{{y=4}}}



This means that the minimum of the range is {{{y=4}}} which tells us that the range is {{{y>=4}}} 



So the range in set-builder notation is 

*[Tex \LARGE \textrm{\left{y|y\ge4\right}}]



and the range in interval notation is 


[4,*[Tex \LARGE \infty])




Now if we graph {{{y=sqrt(x-2)+4}}} , we can visually verify our answer.


{{{ graph( 500, 500, -10, 10, -10, 10, sqrt(x-2)+4) }}}