What is the range for the following function? y=1/(x+2)+3
a){y: y∈ℝ, y≠-2}
b){y: y∈ℝ, y≠2}
c){y: y∈ℝ, y≠3}
d){y: y∈ℝ}

Step-by-step explanation:

The range is the spread of the y-values (minimum to maximum distance travelled).

The graph of your function is a hyperbola shifted two units left and three units up from the origin.

There is a vertical asymptote at x = -2, so y does not exist when x = -2.  However,

Since the limit from either side is the same,

The graph below shows the asymptotes of your function.

Thus. y can take any value  except 3.

In set builder notation, the range is  

f(x) has value 150 at x= 0

and 0 at x= 3

so d(f(x))/dt = slope = -150/3 = -50

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step-by-step explanation: