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.
566 exam-style questions on OCR A Level Maths 1.2 Algebra and Functions, covering 1.2.1 Indices, 1.2.2 Surds, 1.2.3 Simultaneous equations, 1.2.4 Quadratic functions, 1.2.5 Completing the square, 1.2.6 Solving quadratic equations, 1.2.7 Inequalities, 1.2.8 Expressing solutions of inequalities, 1.2.9 Representing inequalities graphically, 1.2.10 Manipulating polynomials, 1.2.11 Simplifying rational expressions, 1.2.12 The modulus function (A-level only), 1.2.13 Graphs of functions, 1.2.14 Sketching curves from equations, 1.2.15 Sketching reciprocal curves, 1.2.16 Interpreting solutions graphically, 1.2.17 Intersection points of graphs, 1.2.18 Proportional relationships, 1.2.19 Graph of the modulus of a linear function (A-level only), 1.2.20 Solving modulus equations graphically (A-level only), 1.2.21 Definition of a function (A-level only), 1.2.22 Inverse and composite functions (A-level only), 1.2.23 Simple graph transformations, 1.2.24 Combinations of graph transformations (A-level only), 1.2.25 Partial fractions (A-level only), and 1.2.26 Models in context. Each one has a worked solution and a mark scheme showing where the marks go.