Question

Explain why the range of f(x)=arcsin(x) is different than g(x)=arccos(x) .

Answer 1

The range of a function is the set of all possible output values of the function. The range of f(x) = arcsin(x) is different than the range of g(x) = arccos(x) because the domain of each function is different.

The function f(x) = arcsin(x) is the inverse sine function, and its domain is the set of all real numbers between -1 and 1, inclusive. This is because the sine function has a range of [-1,1], and the inverse sine function takes the input value from the range of sine function and gives the corresponding angle in radians. So the range of arcsin(x) is [-π/2, π/2]

On the other hand, the function g(x) = arccos(x) is the inverse cosine function, and its domain is the set of all real numbers between -1 and 1, inclusive. This is because the cosine function also has a range of [-1,1], and the inverse cosine function takes the input value from the range of cosine function and gives the corresponding angle in radians. The range of arccos(x) is [0, π]

So, in summary, the difference in the range of f(x) = arcsin(x) and g(x) = arccos(x) is due to the difference in the domain of each function.

Answer 2

Ok this is going to be a bit confusing, and I may have-actually nevermind.

Explanation:

The range of a function is the set of all possible output values that the function can produce. The range of f(x) = arcsin(x) is different than the range of g(x) = arccos(x) because the domain of the two functions is different.

The domain of f(x) = arcsin(x) is -1 <= x <= 1. This means that x can only take on values between -1 and 1 inclusive. The range of f(x) = arcsin(x) is -(pi/2) <= f(x) <= (pi/2). This is because the arcsine function maps the interval [-1, 1] onto the interval [-(pi/2), (pi/2)].

On the other hand, the domain of g(x) = arccos(x) is also -1 <= x <= 1, but the range of g(x) = arccos(x) is 0 <= f(x) <= pi. This is because the arccosine function maps the interval [-1, 1] onto the interval [0, pi].

Both functions are defined for the same values of x, but the range of arcsin(x) is between -(pi/2) and (pi/2) while the range of arccos(x) is between 0 and pi. This means that the range of arcsin(x) is a subset of the range of arccos(x).

Another way to see this is that the arcsin(x) and arccos(x) are inverse functions, meaning that arcsin(x) = y is equivalent to arccos(sin(y)) = x and arccos(x) = y is equivalent to arcsin(cos(y)) = x.

Therefore the range of arcsin(x) will be the set of all possible y such that -1 <= sin(y) <= 1 and the range of arccos(x) will be the set of all possible y such that -1 <= cos(y) <= 1.