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How do you use the vertical line test to show $\sqrt {{x^2} - 4} - y = 0$ is a function?

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Last updated date: 23rd Feb 2024
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Answer
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Hint: In order to solve the above problem first we must know what the vertical line test.Vertical line test is a visual method to determine whether a given curve is a graph of a function or not. A function can have on unique output y with each unique input x.
Using the above given hint we will solve the problem.

Complete step by step answer:
Let's discuss Vertical line methods in more details before performing the calculations.
In vertical line tests we use to plot the graph of the given expression and then we use it to draw a vertical line intersecting the graph on an x-y plane. If the line intersects the graph at more than one point then the given expression is not considered as a function. If the vertical line cuts the graph at only one point then the expression is considered as a function. In short words we can say that for every value of x we must have a unique value of y.
Now, we will plot the graph of the given equation,
We will arrange the terms of the given equation,
$ \Rightarrow y = \sqrt {{x^2} - 4} $ (On squaring both the sides)
$ \Rightarrow {y^2} = {x^2} - 4$ (On shifting on the LHS the x term)
$ \Rightarrow {y^2} - {x^2} = - 4$
or we can write as;
$ \Rightarrow {x^2} - {y^2} = 4$ (We observe that the obtained equation is of rectangular hyperbola)
The general equation of hyperbola is;
${x^2} - {y^2} = {a^2}$
We will draw the graph of the hyperbola on a condition that y>0, so the negative part of the y in x-y plane will not be considered.

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In the graph given above the vertical line intersects the graph at one point only, which means that the given equation is a function.

Note: Instead of using the graphical method we can also use the substitution method to check whether the given expression is a function or not. In the substitution method we use to put different values in the given expression starting from 0,1,2......and so on. If the expression gives different values for each number then the expression is a function otherwise the expression is not the function.