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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.