A graph is the sideways parabola $x=y^2-1$ (opening to the right). Decide whether its inverse $x\leftrightarrow y$ describes a function, and justify using a line test.

Because the graph is a sideways parabola, it automatically yields an inverse that is also a parabola and hence is always a one-to-one function without any need for restriction

The original graph is a function $y=f\left(x\right)$ because it passes the vertical line test, and its inverse is likewise a function by symmetry, so $x\leftrightarrow y$ is a valid function everywhere

The original graph fails the vertical line test if read as y as a function of $x$, but it passes the horizontal line test when viewed as $x=g\left(y\right)$, so its inverse is not a function unless we restrict the domain

The inverse exists and is single-valued for all $x$ because horizontal lines intersect the original graph at most once, guaranteeing invertibility over the whole domain