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What are the connections to word problems for direct and Inverse Variations and why are they functions?

I really don't understand your question. If there is a direct or inverse relationship between two variables, either the word problem will say so or it will be obvious from laws of physics or logic.

it's like how are word problems related to direct or inverse variations?...like when would you use it and how would you use it in a word problem...about the function part...my guess was that whatever output you have, the input will never be the same...is that right?

If you have one output for every input, the output is a function of the input.

Direct variation means that the absolute value of the output goes up with the input.
For example with a spring, the further ou push it, the harder it pushes back.
F = -k x
force varies directly with distance x
or
Potential energy U increases with square of distance
U = (1/2) k x^2
Potential energy again varies directly with distance x

Inverse variation means that the absolute value of output goes down as
the input goes up.
For example with gravity the Force on mass m dues to mass M a distance D away is
F = G m M/D^2
so the force varies inversely with distance and is proportional to 1/distance squared.

however, how is it related to word problems???

Make one up with a spring and make one up with gravity.

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