Determine whether a relation is a function of x

Expert intro: A relation is a function when every input $x$ is paired with exactly one output $y$. The quickest check here is to solve for $y$ and see whether any $x$ can produce two different $y$ values.

Detailed walkthrough

We are given the relation

$x=\pm\sqrt{1-y}$

To test whether it represents $y$ as a function of $x$, solve for $y$.

Square both sides:

$x^2=1-y$

Now isolate $y$:

$y=1-x^2$

At this point, each $x$ gives exactly one $y$ value. That means the relation does represent $y$ as a function of $x$.

You can also notice that the original relation uses $\pm$, which can make it look like there are two outputs. But after rewriting it in terms of $y$, the relation becomes a single-valued rule:

$\boxed{y=1-x^2}$

So the answer is yes, it is a function of $x$.

💡 Pitfall guide

One trap is stopping at $x=\pm\sqrt{1-y}$ and concluding it is not a function because of the plus/minus sign. That sign belongs to the way the relation is written, but once you solve for $y$, the output is unique for each $x$. Another mistake is writing $y=-x^2+1$ with the wrong sign order and then not checking that it matches the same expression.

🔄 Real-world variant

If the equation were instead $y^2=x+1$, then it would not define $y$ as a function of $x$ because most $x$ values would correspond to two possible $y$ values, one positive and one negative. But in this case, after solving, the relation reduces to a single quadratic function.

🔍 Related terms

function, relation, vertical line test