SOLUTION: If the range of g(x) = x^2+4 is all real numbers from 20 to 53, what negative numbers are in the domain of g(x) ? Use interval notation to express your answer.

First we draw the graph of y = g(x) = x2+4

 

Then we draw horizontal lines at y=20 and at y=53, and indicate
(in green on the y-axis) the given range.



Then we find the points of intersection of the
two horizontal lines and g(x), by solving these
two systems:

 and 

by substitution and factoring:

The points of intersection are (-7,53), (-4,20), (4,20), (7,53)





Since the range is ,

we erase the bottom of the graph that drops below y=20,
and also the top of the graph that rises above y=53, for
they are not part of g(x) with the given range:



Now to find the domain, we draw lines from the endpoints
of the graph down to the x-axis, shown below, and indicate 
(in green on the x-axis) the entire domain of g(x) 



The interval for the entire domain (in green on the x-axis) is



However, the question only asks for the NEGATIVE part of the
domain, so we only answer the left part of it, which is just
this left part of the interval notation where x is negative:



Edwin