Question 240654
An equation which expresses y as a function of x is an equation which can be solved for y. This means we can get y by itself on one side:
y = single-valued-expression (without-y's)<br>
Since the first three equations are already in this form, they express y as a function of x. So the only possible answers are (D) or (E). Let's try to solve (D) for y:
{{{x = y^2 + 4}}}
Subtract 4 from each side:
{{{x-4 = y^2}}}
In order to eliminate the exponent we find the square root of each side:
{{{sqrt(x-4) = sqrt(y^2)}}}
In simplifying the right side, a common error is to forget to use absolute value. {{{sqrt(y^2)}}} is not just "y". It is {{{abs(y)}}}. So the equation simplifies to:
{{{sqrt(x-4) = abs(y)}}}
And solving an absolute value equation requires two equations:
{{{sqrt(x-4) = y}}} or {{{-sqrt(x-4) = y}}}
Often some of these steps are skipped and a jump is made from
{{{sqrt(x-4) = sqrt(y^2)}}}
to the shorthand (abbreviation) for the pair of equations:
{{{0 +- sqrt(x-4) = y}}}  (Excuse the extra 0. Algebra.com's formula software can't do "plus-or-minus" without something in front of it.)<br>
Whether we use the two separate equations or the shorthand, we are not able to express y as a "single-valued" function of x. So the answer is (D).<br>An alternate way to solve this kind of problem is to use graphs. If you are good at graphing you can use the vertical line test to see if y is a function of x. (D) works out to be a parabola that opens to the right. If you can picture this you will realize that it fails the vertical line test. (The others all pass.)