Simplification of rational expressions including factorising and cancelling, and algebraic division.

Denominators of rational expressions will be linear or quadratic, eg \(\dfrac{1}{ax + b^{\prime}}\)

\(\dfrac{ax + b}{px^2 + qx + r} , \dfrac{x^3 + 1}{x^2 - 1}\) .

Definition of a function. Domain and range of functions. Composition of functions. Inverse functions and their graphs.

The concept of a function as a one-one or many-one mapping from \(\mathbb{R}\) (or a subset of \(\mathbb{R}\)) to \(\mathbb{R}\). The notation \(f : x\mapsto\) and \(f(x)\) will be used.

Students should know that \(fg\) will mean 'do \(g\) first, then \(f\)'.

Students should know that if \(f^{−1}\) exists, then \(f^{−1}f(x) = ff^{−1}(x) = x\).

The modulus function.

Students should be able to sketch the graphs of \(y = \lvert ax + b \rvert\) and the graphs of \(y = \lvert f(x) \rvert\) and \(y = f(\lvert x \rvert)\), given the graph of \(y = f(x)\).

Combinations of the transformations \(y = f(x)\) as represented by \(y = af(x)\), \(y = f(x) + a\), \(y = f(x + a)\), \(y = f(ax)\).

Students should be able to sketch the graph of, for example, \(y = 2f(3x)\), \(y = f(−x) + 1\), given the graph of \(y = f(x)\) or the graph of, for example, \(y = 3 + \sin 2x\),

\(y = −\cos \left(x + \dfrac{\pi}{4}\right)\).

The graph of \(y = f(ax + b)\) will not be required.

Knowledge of secant, cosecant and cotangent and of arcsin, arccos and arctan. Their relationships to sine, cosine and tangent. Understanding of their graphs and appropriate restricted domains.

Knowledge and use of \(\sec^2 \theta = 1 + \tan^2 \theta\) and \(cosec^2 \theta = 1 + \cot^2 \theta\).

Angles measured in both degrees and radians.

Knowledge and use of double angle formulae; use of formulae for \(\sin (A ± B)\), \(\cos (A ± B)\) and \(\tan (A ± B)\) and of expressions for \(a \cos \theta + b \sin \theta\) in the equivalent forms of \(r \cos (\theta ± a)\) or \(r \sin (\theta ± a)\).

To include application to half angles. Knowledge of the \(t (\tan \frac{1}{2} \theta )\) formulae will not be required.

Students should be able to solve equations such as \(a \cos \theta + b \sin \theta = c\) in a given interval, and to prove simple identities such as \(\cos x \cos 2x + \sin x \sin 2x ≡ \cos x\).

What students need to learn:

The function \(e^x\) and its graph.

To include the graph of \(y = e^{ax + b} + c\).

The function \(\ln x\) and its graph; \(\ln x\) as the inverse function of \(e^x\).

Solution of equations of the form \(e^{ax + b} = p\) and \(\ln (ax + b) = q\) is expected.

What students need to learn:

Differentiation of \(e^x, \ln x, \sin x, \cos x, \tan x\) and their sums and differences.

Differentiation using the product rule, the quotient rule and the chain rule.

Differentiation of \(cosec x, \cot x\) and \(\sec x\) are required. Skill will be expected in the differentiation of functions generated from standard forms using products, quotients and composition, such as \(2x^4 \sin x, \dfrac{e^{3x}}{x}, \cos x^2\) and \(\tan^2 2x\).

The use of \(\dfrac{dy}{dx} = \dfrac{1}{\left(\dfrac{dx}{dy}\right)}\).

Eg finding \(\dfrac{dy}{dx}\) for \(x = \sin 3y\).

What students need to learn:

Location of roots of \(f(x) = 0\) by considering changes of sign of \(f(x)\) in an interval of \(x\) in which \(f(x)\) is continuous.

Approximate solution of equations using simple iterative methods, including recurrence relations of the form \(x_{n+1} = f(x_n)\).

Solution of equations by use of iterative procedures for which leads will be given.