Final answer:
The domain of the function f(x) = 1/(x+3) - 1 is all real numbers except x = -3, and the range is all real numbers except y = -1, due to the presence of a horizontal asymptote at y = -1.
Step-by-step explanation:
Finding the Domain and Range of a Function
The question asks to find the domain and range of the function f(x) = 1/(x+3) - 1. To determine the domain of the function, look for values of x that would make the function undefined. Since division by zero is undefined, we must exclude any x values that result in the denominator being zero.
For this function, setting the denominator x+3 equal to zero gives us x = -3. Thus, every real number except x = -3 is included in the domain of this function.
To find the range of the function, we must understand the behaviour of the function. As x approaches infinity, the function approaches -1 from above, and as x approaches -3 from the left, the function decreases without bound. As x approaches -3 from the right, the function increases without bound.
Therefore, the range of the function is all real numbers except for y = -1, where the horizontal asymptote of the function is found.
In conclusion, the domain of f(x) = 1/(x+3) - 1 is all real numbers except x = -3, and the range is all real numbers except y = -1.