Solving f(x) = k

So far you put an input into a function and got the output. Solving goes the other way. You know the output, and you look for the input.

For $f\left(x\right) = {x}^{2}-1$ and the output 3, write the rule equal to 3: ${x}^{2}-1 = 3$. Add 1 to both sides: ${x}^{2} = 4$. So x = 2 or x = -2. The curve meets the line y = 3 two times.

Two inputs can have the same output. That is fine for a function. But one input can never have two outputs. On a graph, a vertical line never crosses a function two times. So a circle is not a function.