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.
532 exam-style questions on AQA A Level Maths 1.5 B: Algebra and functions, covering 1.5.1 Laws of indices, 1.5.2 Surds, 1.5.3 Quadratic functions, 1.5.4 Simultaneous equations, 1.5.5 Linear and quadratic inequalities, 1.5.6 Polynomials and rational expressions, 1.5.7 Graphs of functions and proportion, 1.5.8 Composite and inverse functions (A-level only), 1.5.9 Transformations of graphs, 1.5.10 Partial fractions (A-level only), and 1.5.11 Functions in modelling (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.