(6) Let \( f ( x ) = 3 - 4 \sin ( 2 x + 1 ) \) and \( g ( x ) = x ^ { 2 } . \) Answer the following question and give a brief justification of your answer. Each part is worth six marks.
(a) What is the domain of
(b) Determine \( \frac { f ( x ) } { g ( x ) - ( g ( x ) ) ^ { 1 / 2 } } ? \)
(c) What is the range of \( f ( x ) ?\)
Find the solution of the exponential equation \( 16 e ^ { x } = 12 \) in terms of logarithms, or correct to four decimal places. \( x = \)
Certain radioactive material decays in such a way that the mass remaining after \( t \) years is given by the function
\( m ( t ) = 100 e ^ { - 0.01 t } \)
where \( m ( t ) \) is measured in grams. (a) Find the mass at time \( t = 0 \) . Your answer is (b) How much of the mass remains after \( 20 \) years? Your answer is Round answers to \( 1 \) decimal place.
