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Section 3-4 : The Definition of a Function

5. Determine if the given equation is a function.

\[y = \sqrt {3{x^2} + 1} \] 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 case we can kind of talk our way through this.

Look at the equation and notice that if we were to plug any \(x\) into the equation all we would do is square the \(x\), multiply that by 3 and then add 1 to that result. For each of these algebraic operations there is exactly one number that results.

Finally, all we do is take the square root of that result. Recalling that square roots can only give positive numbers (i.e. we don’t add on a \( \pm \) when we take the square root).

So, \(y\) can only be a single value regardless of the \(x\) we plug in.

Therefore, this equation is a function.