Question 1027711
I do not believe the problem posted involves complex numbers.
I believe {{{x}}} and {{{g(x)=(c(1-x^2))^(1/2)}}} are meant to be real numbers.
 
We know that to make {{{g(x)}}} a real number we must have
{{{c(1-x^2)>=0}}} , which limits the domain of the function.
We also know something about the range of the function.
We know that it will be {{{g(x))>=0}}} through the domain of {{{g(x)}}} .
So the graph will be a curve that will always have {{{y=g(x)>=0}}} ,
meaning that it will never go below the x-axis,
and there will be a curve  only where {{{x}}} is in the domain of the function.
 
IF YOU HAVE BEEN STUDYING CONIC SECTIONS (SUCH AS ELLIPSES AND HYPERBOLAS):
The graph will have {{{y=(c(1-x^2))^(1/2)}}}<-->{{{y^2=c(1-x^2)}}}<-->{{{y^2=c-cx^2}}} .
Rearranging we get
{{{cx^2+y^2=c}}}
and dividing both sides by {{{c}}} we get
{{{x^2/1^2+y^2/c=1}}}
 
When {{{c<0}}} ,
{{{x^2/1^2+y^2/c=1}}}<-->{{{x^2/1^2-y^2/(-c)=1}}}<-->{{{x^2/1^2-y^2/(sqrt(-c))^2=1}}}
is the equation of a hyperbola.
The graph we get (for {{{y>=0}}} only) looks like the ones below.
{{{graph(300,300,-5,5,-5,35,6sqrt(x^2-1))}}} for {{{c=-36}}} , when {{{x^2-y^2/36=1}}}<-->{{{y=6sqrt(x^2-1)}}} ,
 
{{{graph(300,300,-5,5,-0.5,5,sqrt(x^2-1))}}} for {{{c=-1}}} , when {{{x^2-y^2=1}}}<-->{{{y=sqrt(x^2-1)}}} , and
 
{{{graph(300,300,-5,5,-0.5,2,0.4sqrt(x^2-1))}}} for {{{c=-0.16}}} , when {{{x^2-y^2/0.16=1}}}<-->{{{y=0.4sqrt(x^2-1)}}} .
They all look the same, because they are the same curve stretched vertically to different degrees, but the change in y-axis scale sort of compensates for the stretch.
 
b) When {{{c<0}}} ,
{{{x^2/1^2+y^2/c=1}}}<-->{{{x^2/1^2+y^2/(sqrt(c))^2=1}}}
is the equation of an ellipse (or circle).
The graph we get (for {{{y>=0}}} only) looks like the ones below.
{{{drawing(300,300,-5,5,-1,9,
grid(0),red(arc(0,0,2,12,180,360))
)}}} for {{{c=36}}} , when {{{x^2/1^2+y^2/6^2=1}}} is the equation of an ellipse with a vertical major axis and vertices at (0,6) and (0,-6),
 
{{{drawing(300,300,-1.5,1.5,-1,2,
grid(0),red(arc(0,0,2,2,180,360))
)}}} for {{{c=1}}} , when {{{x^2+y^2=1}}} is the equation of a circle of radius 1, and
 
{{{drawing(300,300,-1.5,1.5,-1,2,
grid(0),red(arc(0,0,2,0.4,180,360))
)}}} for {{{c=0.16}}} , when {{{x^2+y^2/0.4^2=1}}} is the equation of an ellipse with a horizontal major axis and vertices at (0,1) and (0,-1).