Graph `y=1/(x+3)+3` , give the domain and range, and compare to `y=1/x` .

The function `y=a/(x-h)+k` is a transformation of the parent function `y=1/x` . a performs a vertical stretch or compression, h gives a horizontal translation (shift left/right) while k gives a vertical translation (shift up/down).

The range of such a function is `y!=k` . The rational part can never be 0, so the function can never take on the value k.

The domain of the function is `x!=h` as you cannot divide by zero.

There is a horizontal asymptote at y=k and a vertical asymptote at x=h.

For the problem `y=1/(x+3)+3` we have a=1, h=-3, and k=3. Thus the graph will be the graph of `y=1/x` shifted to the left 3 units and up 3 units. There is a horizontal asymptote at y=3 and a vertical asymptote at x=-3.

The graph:

Compare to the graph of `y=1/x` in red.

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