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Trying to understand what the differnce is between a vertical and horizontal line test. If an equation fails the vertical line test, what does that tell you about the graph? If an equation fails the horizontal line test, what does that tell you about the graph?

Any help would be appreciated, Thanks!

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The definition of a function requires that every $x$ value has at most one $y$ value. It doesn't require that every $y$ value has at most one $x$ value though, so the inverse of a function is not necessarily a function itself.

The vertical line test is to check if a curve is a function. If at some point the function crosses a vertical line twice, it's telling you that there are two $y$ values for that $x$ value, so the curve is not a function.

When you find the inverse of a function, you reflect the function by the line $y=x$, or swap the $x$ and $y$ axes. So if you want to check whether a function has an inverse that is also a function, you can check whether every $y$ value has at most one $x$ value, so you use the horizontal line test. If a horizontal line cuts the curve more than once at some point, then the curve doesn't have an inverse function.

So in short, if you have a curve, the vertical line test checks if that curve is a function, and the horizontal line test checks whether the inverse of that curve is a function.

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