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### Section 3-4 : The Definition of a Function

6. Determine if the given equation is a function.

${y^4} - {x^2} = 16$ Show Solution

To directly determine if an equation is a function can be quite difficult at times. What we need to do is show that for each $$x$$ that we plug into the equation we can only get a single $$y$$ out of the equation.

For this equation let’s first rewrite it a little as follows,

${y^4} = {x^2} + 16$

Now, let’s take a look at a specific $$x$$, say $$x = 0$$. If we plug this into the equation we get,

${y^4} = {0^2} + 16 = 16$

Now, at this point we can see that there are two possible $$y$$ values, $$y = - 2$$ or $$y = 2$$ since for both we have,

${\left( { - 2} \right)^4} = 16\hspace{0.25in}\hspace{0.25in}\,\,\,\,\,\,\,{\mbox{and}}\hspace{0.25in}\hspace{0.25in}\hspace{0.25in}{\left( 2 \right)^4} = 16$

So, we’ve found an $$x$$ for which the equation gives two possible $$y$$ values. Note as well that, for this equation, it doesn’t matter which $$x$$ we choose to use we will get the same result.

Therefore, this equation is NOT a function.