The function fff is defined by
f(x)=3x+2x−3,x∈R,x≠3 f(x) = \frac{3x + 2}{x - 3}, \quad x \in \mathbb{R}, x \neq 3 f(x)=x−33x+2,x∈R,x=3(i) Find f−1(x)f^{-1}(x)f−1(x). (a) (ii) Write down an expression for ff(x)ff(x)ff(x).
The function ggg is defined by
g(x)=x2−4x2,x∈R,0≤x≤6 g(x) = \frac{x^2 - 4x}{2}, \quad x \in \mathbb{R}, 0 \leq x \leq 6 g(x)=2x2−4x,x∈R,0≤x≤6(b) (i) Find the range of ggg. (b) (ii) Determine whether ggg has an inverse. Fully justify your answer.
Show that
gf(x)=−3x2+40x+282x2−12x+18 gf(x) = \frac{-3x^2 + 40x + 28}{2x^2 - 12x + 18} gf(x)=2x2−12x+18−3x2+40x+28It can be shown that fgfgfg is defined for a restricted domain of ggg. If the denominator of fg(x)fg(x)fg(x) is given by x2−4x−6x^2 - 4x - 6x2−4x−6, find the value of aaa that must be excluded from the domain 0≤x≤60 \leq x \leq 60≤x≤6 such that fg(x)fg(x)fg(x) is undefined. Fully justify your answer.