The first thing we can do is sketch the function, #x^2 +3 # is just the simple graph #x^2# shifted 3 upward, using our knowledge of trnsformations...

#x^2#:

graph{x^2 [-10, 10, -5, 5]}

#x^2 + 3 #:

graph{x^2+3 [-9.375, 10.625, -1.56, 8.44]}

Hence to find the domain we must consider what values of #x# will give out a real number of #y#, what values the graph is valid for, looking at the graph, we see that all values of #x# gives a value for #y#, hence:

#color(red)("Domain" : x in RR #

As we know #x^2 # gives a vlaue for all values, possitive and negative

Now to consider the range, we just nned to aks what are all the values that #y# can take on, we see the smallest value of ## is # y = 3#, and all the other values of #y# above also can be found..

#color(red)( "Range" : y>=3 ) #