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## Question

If mc021-1.jpg and mc021-2.jpg, what is the range of mc021-3.jpg?

## Answer

, and , we want to find the range of Notice that Whatever value v can possibly produce, that is the Range of v, becomes the range of u.Then whatever u produces for these values, is the range of u(v(x))So, first find range of v: clearly the domain of v is R-{0}, as v(0) makes no sense.We check what values c can v produce:,this means that any c (for now) can be produced… it is enough to let x=1/cthis also means that c cannot be equal to 0 as 1/c makes no sense if c=0.Thus the range of v is R-{0}, Now we check the range of u(v(x)) for v(x)∈R-{0}, assume we want to produce a value c, so: since the left side is always positive or 0, for x=0, the right hand side expression must also be positive or zero, which means c-3 must be negative or 0, thus c-3≤0; c≤3. Here recall that x in u(x) cannot be 0, so c<3.Answer: The range of u(v(x)) is (-infinity, 3)

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