The function `f` is defined by `f(x)=\sin (x)` for all real numbers `x`. Let `g` be the function defined on `[-1,1]` by `g(x)=\arcsin (x)`.
Is `g` the inverse of `f`? Justify your answer. (2 marks)
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The function `f` is defined by `f(x)=\sin (x)` for all real numbers `x`. Let `g` be the function defined on `[-1,1]` by `g(x)=\arcsin (x)`.
Is `g` the inverse of `f`? Justify your answer. (2 marks)
`text{The domain of}\ \ f(x) = RR`
`text{The range of}\ \ g(x) = [-pi/2,pi/2]`
`text{S}text{ince the domain}\ \ f(x) !=\ text{range of}\ \ g(x),`
`f^(-1)(x)!=g(x)`
`text{The domain of}\ \ f(x) = RR`
`text{The range of}\ \ g(x) = [-pi/2,pi/2]`
`text{S}text{ince the domain}\ \ f(x) !=\ text{range of}\ \ g(x),`
`f^(-1)(x)!=g(x)`
State the domain and range of `y = cos^-1 (x/4).` (2 marks)
--- 5 WORK AREA LINES (style=lined) ---
`text(Domain)\ \ -4 <= x <= 4`
`text(Range)\ \ 0 <= y <= pi`
`y = cos^-1\ x/4`
`text(Domain of)\ \ y =cos^-1 x\ \ text(is)`
`-1 <= x <= 1`
`:.\ text(Domain of)\ \ y = cos^-1\ x/4\ \ text(is)`
`-1 <= x/4 <= 1`
`-4 <= x <= 4`
`text(Range)\ \y = cos^-1\ x\ \ text(is)`
`0 <= y <= pi`
`:.\ text(Range)\ \ y = cos^-1\ x/4\ \ text(is)`
`0 <= y <= pi`
What is the domain of the function `f(x) = sin^(−1)\ (2x)`?
(A) `−pi ≤ x ≤ pi`
(B) `−2 ≤ x ≤ 2`
(C) `−pi/4 ≤ x ≤ pi/4`
(D) `−1/2 ≤ x ≤ 1/2`
`D`
`f(x)= sin^(−1)\ (2x)`
`text(Domain of)\ f(x) = sin^(−1) x\ \ text(is)`
`−1 ≤ x ≤ 1`
`:.text(Domain of)\ \ f(x) = sin^(−1)\ (2x)\ \ text(is)`
`−1 ≤ 2x ≤ 1`
`−1/2 ≤ x ≤ 1/2`
`⇒ D`
Let `f(x) = cos^(-1) (x/2)`. What is the domain of `f(x)`? (1 mark)
`-2 <= x <= 2`
`f(x) = cos^(-1) (x/2)`
`–1` | `<= x/2` | `<= 1` |
`–2` | `<= x` | `<= 2` |
`:.\ text(Domain of)\ \ f(x)\ \ text(is)\ \ \ –2 <= x <= 2`