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How do you decide whether the relation #x - 3y = 2# defines a function?

Answer 1

#f(x)=y=-1/3x+2/3# is a function since for #AA x in RR #
#f(x): RR_x -> RR_y# uniquely.

The take home lesson here is that a #f(x)# is a function if and only if for every value of #x# from a set #X# #f(x)# generates a unique mapping of #y=f(x)# in a set #Y#. i.e. #x# results in a unique single #y#.

So we need to show that the function in your question uniquely maps #f(x): X -> Y, AA x in X=RR#

1) First write you equation in the form: #y=f(x)# #y=f(x)=1/3(2-x)= -1/3x+2/3# Now we see that for every #x in RR# #f(x)# maps #x# to a unique value of #y#, given by #y=-1/3x+2/3#.

So we conclude that #x-3y=2# represented by #f(x)= y=-1/3x+2/3# is indeed a function.

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Answer 2

Genesis Allen

To determine if the relation x - 3y = 2 defines a function, we need to check if each input value (x) corresponds to exactly one output value (y). We can rewrite the equation as y = (x - 2)/3 and observe that for every value of x, there is a unique corresponding value of y. Therefore, the relation defines a function.

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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.