Determine the domain and range of the function f(x) = x / x^ - 1

Would appreciate step by step explanation

It looks like the function is supposed to be

f(x) = x / (x ⁻¹)

which can be rewritten as

f(x) = x / (1 / x)

Without any further simplification, notice that x = 0 makes the denominator undefined, so x = 0 is not in the domain. For all other real numbers, we have

x / (1 / x) = x ²

which exists for all real numbers. So the domain (in interval notation) is (-∞, 0) ∪ (0, ∞). If you want this in set notation, it's the set of all real numbers except 0, which can be written as ℝ\ {0} or ℝ- {0}. Or in set-builder notation, {x ∈ ℝ | x ≠ 0}.

For all x ≠ 0, since the function reduces to x ², it will take on all positive real numbers. Since x ² ≥ 0 under normal circumstances, check to see if f(x) also attains this value:

x / (1 / x) = 0

In order for this to be 0, the numerator must be equal to 0. But if x = 0, then the denominator is undefined. So 0 is not in the range, and the range is (0, ∞), or ℝ⁺ \ {0} (where ℝ⁺ denotes the set of positive real numbers), or {x ∈ ℝ | x > 0}.

maximum area = xy = (16)(32) = 512 ft2

to maximize the area of the garden, the side of the fence parallel to the side of the house should be 32 ft long, and the other two sides should both have length 16 ft.'

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