(Solved): A typical logarithmic function is given by: \[ f(x)=3 \log _{2}(10)+\log _{2}\left(\frac{2+x}{3}\ri ...

A typical logarithmic function is given by: \[ f(x)=3 \log _{2}(10)+\log _{2}\left(\frac{2+x}{3}\right) \] a) Find the equation for its inverse \( f^{-1}(x) \) and specify its implied domain and range. (1 mark) b) Sketch both \( f(x) \) and \( f^{-1}(x) \) on the same set of axes showing all important features. What did you notice about these two graphs? (1 mark) [Note1: Recall a sketch does not use technology (e.g., Excel or graphing software) when being generated! So, draw your plot free hand on a graph paper (you can use tools to see the graphs but you need to present hand sketch). This rule applies to all questions requiring sketch] [Note 2: you need to show calculations how you found these features. This rule applies to all questions that require to specify important features on a graph] Question 2 ( 2 marks) a) Use the exponential definition of \( \sinh (x) \) and \( \cosh (x) \) to show \( \cosh (2 x)= \) \( 2 \cosh ^{2}(x)-1 \). (1 mark) b) Solve the equation \( 4 \cosh (2 x)-2 \sinh (x)=7 \), giving your answer in terms of natural logarithm. (1 mark) Question 3 ( 2 marks) a) Evaluate and simply: \( \cos \left(\frac{\pi}{6}-x\right)-\cos \left(\frac{\pi}{6}+x\right) \) (1 mark) b) Find the value of \( m \) if \( \cot (2 \theta)+\frac{2}{2} \tan (\theta)=m \cot (\theta) \) (1 mark) Question 4 ( 2 marks) Solve the following trigonometric equations: a) \( 2 \cos ^{2}(\beta)-\cos ^{2}(\beta)-2 \cos (\beta)+1=0 \) for \( -2 \pi \leq \beta \leq 2 \pi \quad \) (1 mark) b) \( 2 \operatorname{cosec}\left(3 \theta-\frac{\pi}{3}\right)=\frac{4}{\sqrt{3}} \) for \( -\pi \leq \theta \leq 2 \pi \quad \) (1 mark)