Algebra and Functions

• Understand the terms: function, domain, range (image set), one-one function, many-one function, inverse function and composition of functions. • Includes explaining in words why a given relation is a function.

• Find the domain and range of functions, including inverse and composite functions. • Restrict the domain of f where needed for f^-1 and/or gf to exist. • Understand that the domain of gf is a subset of the domain of f, and the range of gf is a subset of the range of g.

• Recognise and use function notation such as f(x), f: x -> lg x, f^-1(x), fg(x) = f(g(x)) and f^2(x) = f(f(x)). • The notation f^2(x) will not be used with trigonometric functions.

• Understand the relationship between y = f(x) and y = |f(x)| where f(x) may be linear, quadratic, cubic or trigonometric. • For trigonometric functions, use y = a sin bx + c, y = a cos bx + c or y = a tan bx + c, where a is a positive integer, b is a simple fraction or integer, and c is an integer. • Fractions for b have denominator 2, 3, 4, 6 or 8 only.

• Explain in words why a given function does not have an inverse.

• Find the inverse of a one-one function. • Use correct inverse-function notation.

• Form and use composite functions. • Understand that the order of functions is important, so fg may not be the same as gf.

• Use sketch graphs to show the relationship between a function and its inverse. • Understand that each graph is the reflection of the other in the line y = x.

• Find the maximum or minimum value of the quadratic function f: x -> ax^2 + bx + c by completing the square or by differentiation.

• Use the maximum or minimum value of f(x) to sketch y = f(x) or determine the range for a given domain. • Use correct notation to write a domain or range.

• Know the conditions for f(x) = 0 to have two real roots, two equal roots or no real roots. • Relate the discriminant to the roots of the equation. • Apply the related conditions for a line to intersect a curve, be tangent to a curve or not intersect a curve.

• Solve quadratic equations for real roots. • Use factorisation, the quadratic formula and completing the square. • The quadratic formula is given in the List of formulas. • On the calculator paper, correct answers are acceptable without working.

• Find the solution set for quadratic inequalities graphically or algebraically. • Write solutions in the correct form, such as -3 < x < 4 or x < 1 or x > 6.

• Know and use the remainder theorem and the factor theorem.

• Find factors of polynomials. • For a cubic polynomial, first obtain a product of a linear factor and a quadratic factor, for example by observation or algebraic long division.

• Solve cubic equations.

4.1. Solve modulus equations

• Solve equations of the type |ax + b| = c, |ax + b| = cx + d, |ax + b| = |cx + d| and |ax^2 + bx + c| = d using algebraic or graphical methods. • For graphical solutions, draw an accurate graph. • For algebraic methods, any valid method is acceptable.

4.2. Solve modulus inequalities

• Solve graphically or algebraically inequalities involving k|ax + b| and |ax^2 + bx + c|, including comparisons with constants, linear expressions and other modulus expressions. • Use k > 0 and c >= 0 or c > 0 where specified. • For graphical solutions, draw an accurate graph; for algebraic methods, any valid method is acceptable.

4.3. Use substitution for related equations

• Use substitution to form and solve a quadratic equation in order to solve a related equation. • Identify the appropriate substitution in equations involving powers, logarithms or exponentials.

4.4. Sketch cubic polynomials and moduli

• Sketch graphs of cubic polynomials and their moduli when given as a product of three linear factors. • Clearly label points of intersection with the coordinate axes.

4.5. Solve cubic inequalities graphically

• Solve graphically cubic inequalities of the form f(x) >= d, f(x) > d, f(x) <= d and f(x) < d where f(x) is a product of three linear factors and d is a constant.

• Solve simultaneous equations in two unknowns by elimination or substitution. • Includes equations such as a line with a quadratic relation, equations involving xy, and fractional equations.

• Know and use simple properties and graphs of logarithmic and exponential functions, including ln x and e^x. • Understand that f(x) = e^x and g(x) = ln x are inverse functions. • Understand the asymptotic nature of logarithmic and exponential graphs and state equations of asymptotes. • Graphs are limited to y = ke^(nx) + a and y = k ln(ax + b), where n, k, a and b are integers. • Series expansions are not required.

• Know and use the laws of logarithms, including change of base of logarithms. • Combine logarithmic expressions and convert logarithms between bases, including natural logarithms.

• Solve equations of the form a^x = b.