Testes de linha

A function satisfies that for all x in the domain of f, $f(x)=y$. There is only one unique number y, i.e. we cannot have $f(3)=2, 4$ at the same time.

To test for whether something is a function or not, we use the vertical line test.

If there is a vertical line that intersects the graph of the "function" at two different spots, it is not a function.

In our example, $f(x)=x \pm 3$. Now the vertical line $x=0$ intersects $f(x)$ at $(0,3) (0,-3)$ so the thing is not a function.

A function only has an inverse if there is a one-to-one correspondence between $f(x)$ and $x$.

Remember that inverses of functions satisfy $f^{-1} (y)=x$

The ** horizontal line test ** tells us that if a horizontal line intersects a function at two different spots, the function has no inverse. The reason is that if the line hits the graph at $(a, y) (b,y)$ then the inverse function would satisfy $g(y)=a, g(y)=b$, which is impossible.