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

1 Answer
Jun 28, 2018

Answer:

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.

desmos.com/calculator

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.)