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Trigonometry, EXT1 T1 2024 HSC 4 MC

What are the domain and range of the function  \(y = 2 \cos^{-1}(2x) + 2 \sin^{-1}(2x)\)?

  1. Domain: \([-0.5, 0.5]\) and Range: \(\{\pi\}\)
  2. Domain: \([-0.5, 0.5]\) and Range: \([-\pi, 3 \pi ]\)
  3. Domain: \([-2, 2]\) and Range: \(\{\pi\}\)
  4. Domain: \([-2, 2]\) and Range: \([-\pi, 3\pi]\)
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\(A\)

Show Worked Solution

\(\text{Domain:}\ \ -1 \leqslant 2x \leqslant 1 \ \ \Rightarrow\ \ -\dfrac{1}{2} \leqslant x \leqslant \dfrac{1}{2} \)

\(\text{Range:}\ \ 2\Big(\cos^{-1}(2x)+ \sin^{-1}(2x)\Big) = 2 \times \dfrac{\pi}{2} = \pi\)

\(\Rightarrow A\)

Trigonometry, EXT1 T1 EQ-Bank 5

  1. State the domain and range for  \(f(x)=4 \sin ^{-1}(2 x-3)\).   (2 marks)

  2. Sketch the function  \(f(x)=4 \sin ^{-1}(2 x-3)\).  (1 mark)

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a.    \(\text{Domain:}\ \ -1 \leqslant\) \(2 x-3 \leqslant 1 \)
  \( 2 \leqslant\) \(2 x \leqslant 4\)
  \( 1 \leqslant\)  \(x \leqslant 2\)

 

        \(\text{Range:}\ \ -\dfrac{\pi}{2} \leqslant\) \(y \leqslant \dfrac{\pi}{2}\)
  \( -2 \pi \leqslant \) \(4 y \leqslant 2 \pi\)

 
b.

Show Worked Solution

a.    \(\text{Domain:}\ \ -1 \leqslant\) \(2 x-3 \leqslant 1 \)
  \( 2 \leqslant\) \(2 x \leqslant 4\)
  \( 1 \leqslant\)  \(x \leqslant 2\)

 

        \(\text{Range:}\ \ -\dfrac{\pi}{2} \leqslant\) \(y \leqslant \dfrac{\pi}{2}\)
  \( -2 \pi \leqslant \) \(4 y \leqslant 2 \pi\)

 
b.

Trigonometry, EXT1 T1 EQ-Bank 2

State the domain and range of the function

\(y=\arccos \, 3x\)   (2 marks)

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\(\text{Domain:}\ \ \Big[-\dfrac{1}{3}, \dfrac{1}{3}\Big] \)

\(\text{Range:}\ \ [0, \pi] \)

Show Worked Solution

\(\text{Consider domain:}\ \ -1 \leqslant 3x \leqslant 1 \ \Rightarrow\ -\dfrac{1}{3} \leqslant x \leqslant \dfrac{1}{3}\)

\(\text{Domain:}\ \ \Big[-\dfrac{1}{3}, \dfrac{1}{3}\Big] \)

\(\text{Range:}\ \ [0, \pi] \)

Trigonometry, EXT1 T1 2022 HSC 13c


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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`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)`

Show Worked Solution

`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)`


♦♦ Mean mark 24%.

Trigonometry, EXT1 T1 2005 HSC 1c

State the domain and range of  `y = cos^-1 (x/4).`   (2 marks)

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`text(Domain)\ \ -4 <= x <= 4`

`text(Range)\ \ 0 <= y <= pi`

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 `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`

Trigonometry, EXT1 T1 2015 HSC 6 MC

What is the domain of the function `f(x) = sin^(-1)\ (2x)`?

  1. `-pi ≤ x ≤ pi`
  2. `-2 ≤ x ≤ 2`
  3. `-pi/4 ≤ x ≤ pi/4`
  4. `-1/2 ≤ x ≤ 1/2`
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`D`

Show Worked Solution

`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`