Question

How do you use the vertical line test to show `sqrt(x^2-4)-y=0` is a function?

Answer 1

intersects with a vertical line only once

Explanation:

first, rearrange the function so that `y` is by itself on one side.

`sqrt(x^2-4) - y = 0`

`sqrt(x^2-4) = y`

`y = sqrt(x^2-4)`

the graph should look like this:
graph{sqrt(x^2-4) [-10, 10, -5, 5]}

then, pick any number outside the range `-2 < x < 2`, so that you have an `x`-value for which `sqrt(x^2-4)` is defined.

example: `x = -5`
this is a vertical line, all points on which have an `x`-value of `-5`.
in the vertical line test, a graph is shown to be a function if it meets a given vertical line only once.

the image above shows that the graph `y = sqrt(x^2-4)` intersects with the line `x = -5` only once.

(the vertical line test can be done again with other `x`-values, but to show how the test works on a function like this, only one test is sufficient.)