The concentration of a chemical CCC in a reaction vessel at time ttt minutes is modelled by the function:
C(t)=20−4tfor t≤5C(t) = \sqrt{20 - 4t} \quad \text{for } t \le 5C(t)=20−4tfor t≤5
The rate of reaction RRR relative to the concentration is given by:
R(C)=2Cfor C≠0R(C) = \frac{2}{C} \quad \text{for } C \ne 0R(C)=C2for C=0
The overall system function HHH is defined by the composite function H(t)=R(C(t))H(t) = R(C(t))H(t)=R(C(t)) on the maximum possible domain.
Find an expression for H(t)H(t)H(t).
Find the domain of HHH.
Show that H−1(x)=5−1x2H^{-1}(x) = 5 - \frac{1}{x^2}H−1(x)=5−x21.