4 Functions

The idea of a function of a variable

Function as a mapping or as a correspondence between the elements of two sets

Use function notation in the forms f(x) = … and f: x ↦ … .

Domain and range of a function Questions will not be set on continuity, but students will be expected to recognise when parts of the domain need to be excluded (e.g. x = 0 must be excluded from the domain of the function f where f(x) = 1/x)

Composite functions ‘fg’ will mean ‘do g first then f’

Inverse functions Finding the inverse of a function

Variation, direct and indirect proportion To include only the following: y ∝ x; y ∝ 1/x; y ∝ x²; y ∝ 1/x²; y ∝ x³; y ∝ 1/x³; y ∝ √x; y ∝ 1/√x

Rectangular Cartesian co-ordinates

Recognise that equations of the form y = mx + c are straight–line graphs with gradient m and intercept on the y-axis at the point (0, c)

Graphs and graphical treatment of the equation: y = Ax³ + Bx² + Cx + D + E/x + F/x², in which the constants are numerical and at least three of them are zero Students will be expected to draw and interpret graphs from given equations Use of the intersection of two curves (graphs) to solve equations

The gradients of graphs above by drawing Students will be expected to draw a reasonable tangent to the graph at a named point and to construct an appropriate right-angled triangle from which to calculate the gradient

Differentiation of integer powers of x Use of dy/dx notation

Determination of gradients, rates of change, maxima and minima, stationary points and turning points Students will either be required to differentiate or use graphical methods to arrive at solutions and relate their calculations to their graphs and vice versa

Applications to linear kinematics and to other simple practical problems This includes the drawing and interpretation of distance/time and speed/time graphs, and other graphs of a similar nature Students need to be able to understand the relationship between displacement or distance, velocity and speed, and acceleration, for example: ds/dt = v and dv/dt = a