My explanation to these questions:
First of all, as per the graph it seen that a curve represents a function if drawing a vertical line to
the graph, this graph cuts it at the single point.
Does this equation determine a relation between x and y?
As identified in introduction of the problem, X and Y are related because this is a relationship
that in the Cartesian plane is represented by circumstances of center (0,2) and radius 1.
Can the variable x can be seen as a function of y, like x=g(y)?
Yes, it could be. By clearing y as a function of X or an X as function of Y. let me explain it
mathematically,
x2+(y−2) ^2=1 → (y−2)2=1 -x2→y−2=±1 -x2→y=2±1-x2 →
y1=2+√ 1-x2
y2=2−√ 1-x2 Each of these branches separately defines a function.
Can the variable y be expressed as a function of x, like y= h(x)?
Well, here the domain of y1 and y2 are the same and are obtained by making the sub radical
amount equal to zero or greater than zero, that’s
1 – x^2 ≥ 0 (1 – x) (1 +x) ≥ 0, so the domain will be [-1,1]
If these are possible, then what will be the domains for these two functions?
The equation to be solved is x^2 (y-4) ^2 =1
Add -y^2 to both sides.
x2+y2−8y+16+−y2=1+−y2
xy2−8y+16=−y2+1
Add 8y to both sides.
x2−8y+16+8y=−y2+1+8y
x2+16=−y2+8y+1
Add -16 to both sides.
x2+16+−16=−y2+8y+1+−16
x2=−y2+8y−15
then Take square root.
x=√−y2+8y−15 or x=−√−y2+8y−15
First of all, as per the graph it seen that a curve represents a function if drawing a vertical line to
the graph, this graph cuts it at the single point.
Does this equation determine a relation between x and y?
As identified in introduction of the problem, X and Y are related because this is a relationship
that in the Cartesian plane is represented by circumstances of center (0,2) and radius 1.
Can the variable x can be seen as a function of y, like x=g(y)?
Yes, it could be. By clearing y as a function of X or an X as function of Y. let me explain it
mathematically,
x2+(y−2) ^2=1 → (y−2)2=1 -x2→y−2=±1 -x2→y=2±1-x2 →
y1=2+√ 1-x2
y2=2−√ 1-x2 Each of these branches separately defines a function.
Can the variable y be expressed as a function of x, like y= h(x)?
Well, here the domain of y1 and y2 are the same and are obtained by making the sub radical
amount equal to zero or greater than zero, that’s
1 – x^2 ≥ 0 (1 – x) (1 +x) ≥ 0, so the domain will be [-1,1]
If these are possible, then what will be the domains for these two functions?
The equation to be solved is x^2 (y-4) ^2 =1
Add -y^2 to both sides.
x2+y2−8y+16+−y2=1+−y2
xy2−8y+16=−y2+1
Add 8y to both sides.
x2−8y+16+8y=−y2+1+8y
x2+16=−y2+8y+1
Add -16 to both sides.
x2+16+−16=−y2+8y+1+−16
x2=−y2+8y−15
then Take square root.
x=√−y2+8y−15 or x=−√−y2+8y−15
