SOLUTION: For Each of the following, determine whether the equation defines "y" as a function of "x". x^2+|y|=9 3x=y^2 y=|x+8| x^2+8y=8

x² + |y| = 9
     |y| = 9 - x²

Make two equations

y = 9 - x²      y = -(9 - x²)

Graph them both on the same set of axes:



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



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.

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3x = y²

y² = 3x

Use the principle of square roots:
      __
y = ±Ö3x 

Make two equations:

     __        __
y = Ö3x, y = -Ö3x


Graph them both on the same set of axes:



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



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:



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





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 - x²

Draw that graph:



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





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