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

Jun 28, 2018

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.)
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