Which equation is a function? $y^{2}=7+x^{2}$ $y=x^{2}-9 x+5$ $x=y^{2}-8$ $x=|y|$

Step 1 ：A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. This is also known as the vertical line test.

Step 2 ：Looking at the equations, we can see that the first equation \(y^{2}=7+x^{2}\) is not a function because for a given x-value, there can be two possible y-values (one positive and one negative).

Step 3 ：The third equation \(x=y^{2}-8\) is also not a function because for a given y-value, there can be two possible x-values (one positive and one negative).

Step 4 ：The fourth equation \(x=|y|\) is also not a function because for a given y-value, there can be two possible x-values (one positive and one negative).

Step 5 ：The second equation \(y=x^{2}-9 x+5\) is a function because for every x-value there is exactly one y-value.

Step 6 ：Final Answer: The equation that is a function is \(\boxed{y=x^{2}-9 x+5}\).