i) What is the domain of the function f(x) = arccos (|x-1|)?
ii) Simplify the function sin(arcsin(x))
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i) Arccos is the inverse cosine function. Since the range (the range of y values) of the cosine function is [-1,1] then the domain of the inverse cosine function, arccos, is the same. With an inverse function we simply swap the domain and range over.
If we let `f(y) = arccos(y)` be a standard arccos function, then if `y= |x-1|` then `y` can only take positive values. Therefore `y in [0,1]` rather than the full range [-1,1].
Further, `y = |x-1|` so that `x-1 = pm y` implying that
`x = 1 pm y`
Therefore `x` is in [1,2] and the domain of the function is then [1,2].
ii) Since arcsin is the inverse function of the sine function then
` `sin(arcsin(x)) is simply equal to `x` (we carry out the operation on `x` and then carry out the reverse operation on the result, giving us back the value `x`).
i) The domain of arccos(|x-1|) is x in [1,2]
ii) sin(arcsin(x)) is simply x
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