The domain of the composite function (g ∘ f)(x), given f(x) = 0.2 / (x^2 - 1) and g(x) = x + 1, excludes -1 and 1, and the range is all real numbers except 1.
The task is to determine the domain and range of (g ∘ f)(x) given that f(x) = 0.2 / (x^2 - 1) and g(x) = x + 1.
To find the domain of the composite function, we must consider the domains of both f(x) and g(x).
First, the domain of f(x) is all real numbers except where the denominator equals zero.
Since the denominator is x^2 - 1, the values that make it zero are x = 1 and x = -1 (because (1)^2 - 1 = 0 and (-1)^2 - 1 = 0).
Therefore, the domain of f(x) excludes -1 and 1.
Next, for the composite function (g ∘ f)(x), we must apply the output of f(x) to g(x).
Since g(x) simply adds 1 to its input and has no further restrictions, the domain of (g ∘ f) will be the same as the domain of f(x).
The range of f(x) is all real numbers because it's a rational function with a hyperbola shape, but for (g ∘ f)(x), we need to consider that each y-value is increased by 1.
The values that are not in the range of f(x) are y=0 when x approaches ±1, which become y=1 in g(x).
So, y=1 will not be in the range of (g ∘ f)(x). Hence, the range is all real numbers except 1.
The correct option is B. Domain: X not equal-1, 1 and Range: (-∞, -1) U (1, ∞).
The probable question may be:
Determine the domain and range of (g o f)(x) if f (x)=.2/x^2-1 and g(x)=x+1
A. D: X not equal-2, 0 and R: (-∞,-2) U (0, ∞)
B. D: (X∈R| X not equal-1, 1} and R: (-∞, -1) U (1, ∞)
C. D: (X∈R| X not equal -2, -1, 0, 1} and R: (-∞, -2) U (0, ∞)
D. D: {X∈R) and R: (-∞, -1) U (1,∞)