Question

(b) find the domain d and range r of f.
domain:
d = (-∞, 0
d = all real numbers
d = (-∞, 0) ∩ (0, ∞)
d = (-∞, 0) ∪ (0, ∞)
d = (-∞, 8) ∪ (8, ∞)
range:
r = (-∞, 8) ∪ (8, ∞)
r = (-∞, 0) ∪ (0, ∞)
r = 0, ∞)
r = all real numbers
r = (-∞, 0) ∩ (0, ∞)
(c) find the intervals on which f is increasing or is decreasing.
decreasing on ℝ
decreasing on (-∞, 0) and increasing on (0, ∞)
increasing on ℝ
increasing on (-∞, 0) and decreasing on (0, ∞)
decreasing on (-∞, 0) and decreasing on (0, ∞)

Explanation:

Step1: Recall domain definition

The domain of a function is the set of all possible input - values (x - values). Without knowing the function \(f\), if there is a vertical asymptote or a point of discontinuity at \(x = 0\), the domain is all real numbers except \(x=0\), which is written as \(D=(-\infty,0)\cup(0,\infty)\).

Step2: Recall range definition

If the function has a horizontal asymptote or some non - allowed output values, and no information about the function suggests non - negative values only or values around \(8\) are excluded, for a rational or some other common non - bounded functions, if there are no restrictions on the output based on the domain and the nature of the function, the range is all real numbers \(R = \text{all real numbers}\).

Step3: Recall increasing and decreasing intervals

If the function has a vertical asymptote at \(x = 0\), and for example, if it is a hyperbola - like function \(y=\frac{1}{x}\), it is decreasing on \((-\infty,0)\) and decreasing on \((0,\infty)\).

Answer:

Domain: \(D = (-\infty,0)\cup(0,\infty)\)
Range: \(R=\text{all real numbers}\)
Increasing/Decreasing: decreasing on \((-\infty,0)\) and decreasing on \((0,\infty)\)