Question 393134
<pre><font face = "batangche" color = "indigo" size = 4><b>
x² + |y| = 9
     |y| = 9 - x²

Make two equations

y = 9 - x²      y = -(9 - x²)

Graph them both on the same set of axes:

{{{drawing(200,400,-6,6,-12,12,


graph(200,400,-6,6,-12,12,9-x^2),
graph(200,400,-6,6,-12,12,-9+x^2)
)}}}

Let's pass some green vertical lines through the graph
to see if it passes the vertical line test:

{{{drawing(200,400,-6,6,-12,12,
green(line(-4,13,-4,-13),
line(-2,13,-2,-13),
line(-6,13,-6,-13),
line(-1,13,-1,-13),
line(2.3,13,2.3,-13),
line(4.5,13,4.5,-13),
line(5.5,13,5.5,-13)),

graph(200,400,-6,6,-12,12,9-x^2),
graph(200,400,-6,6,-12,12,-9+x^2)
)}}}

No, it does not define y as a function of x because many of those green
vertical lines intersect the graph twice.  In order for y to be defined as a
function of x, the graph must be such that no vertical line ever crosses the
graph but once.

-----------------------

3x = y²

y² = 3x

Use the principle of square roots:
      __
y = ±<font face = "symbol">Ö</font>3x 

Make two equations:

     __        __
y = <font face = "symbol">Ö</font>3x, y = -<font face = "symbol">Ö</font>3x


Graph them both on the same set of axes:

{{{drawing(400,400,-6,6,-6,6,


graph(400,400,-6,6,-6,6,sqrt(3x)),
graph(400,400,-6,6,-6,6,-sqrt(3x))
)}}}

Let's pass some green vertical lines through the graph
to see if it passes the vertical line test:

{{{drawing(400,400,-6,6,-6,6,
green(line(-4,13,-4,-13),
line(-2,13,-2,-13),
line(-6,13,-6,-13),
line(-1,13,-1,-13),
line(2.3,13,2.3,-13),
line(4.5,13,4.5,-13),
line(5.5,13,5.5,-13)),




graph(400,400,-6,6,-6,6,sqrt(3x)),
graph(400,400,-6,6,-6,6,-sqrt(3x))
)}}}

No, it does not define y as a function of x because some of those green
vertical lines intersect the graph twice.  In order for y to be defined as a
function of x, the graph must be such that no vertical line ever crosses the
graph but once.

-----------------------
y = |x + 8|

Make two equations

y = x + 8      y = -(x + 8)

Since absolute values are never negative, we do not
use any points on either graph which are below the
x-axis:


Graph them both on the same set of axes, but do not
extend either graph below the x-axis:

{{{drawing(400,400,-20,4,-12,12,


graph(400,400,-20,4,-12,12,abs(x+8))
)}}}

Let's pass some green vertical lines through the graph
to see if it passes the vertical line test:
{{{drawing(400,400,-20,4,-12,12,

green(line(-4,13,-4,-13),
line(-2,13,-2,-13),
line(-6,13,-6,-13),
line(-1,13,-1,-13),
line(2.3,13,2.3,-13),
line(4.5,13,4.5,-13),
line(5.5,13,5.5,-13),
line(-15,13,-15,-13),
line(-10,13,-10,-13)), 
graph(400,400,-20,4,-12,12,abs(x+8))
)}}}




Yes, it does define y as a function of x because none of those green
vertical lines intersect the graph twice, but only once.

-----------------------

x² + 8y = 8
     8y = 8 - x²
      y = 1 - {{{1/8}}}x²

Draw that graph:

{{{drawing(400,400,-10,10,-16,4,


graph(400,400,-10,10,-16,4,1-(1/8)x^2)
)}}}

Let's pass some green vertical lines through the graph
to see if it passes the vertical line test:
{{{drawing(400,400,-10,10,-16,4,

green(line(-4,13,-4,-17),
line(-2,13,-2,-17),
line(-6,13,-6,-17),
line(-1,13,-1,-17),
line(2.3,13,2.3,-17),
line(4.5,13,4.5,-17),
line(5.5,13,5.5,-17),
line(-15,13,-15,-17),
line(-10,13,-10,-17)), 
graph(400,400,-10,10,-10,10,1-x^2/8)
)}}}




Yes, it does define y as a function of x because none of those green
vertical lines intersect the graph twice, but only once.

-----------------------
Edwin</pre>