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How do you decide whether the relation #xy=9# defines a function?

Answer 1

Function means: at most one #y# for every #x#

We can rewrite as #y=9/x# or even #x=9/y#

In both cases we get only one answer, except when #x=0# or #y=0# so we can say: #y=f(x)# and #x=g(y)# graph{9/x [-22.8, 22.83, -11.4, 11.4]}

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Answer 2

Kennedy Adams

To determine if the relation (xy = 9) defines a function, we can analyze whether each input value (x) corresponds to exactly one output value (y).

If we solve the equation (xy = 9) for (y), we get (y = \frac{9}{x}). This implies that for any value of (x) except (x = 0), there will be a corresponding value of (y). However, for (x = 0), (y) becomes undefined, which means there is no unique output for (x = 0).

Therefore, the relation (xy = 9) does not define a function because it fails the vertical line test: there exists at least one vertical line ((x = 0)) that intersects the graph of the relation at more than one point.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.