#4A
The idea of a function of a variable
#4B
Function as a mapping or as a correspondence between the elements of two sets
#4C
Use functional notations of the form \(f(x) =...\) and \(f : x ↦ ...\)
#4D
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 \(f(x) = \dfrac{1}{x}\))
#4E
Composite functions
‘\(fg\)’ will mean ‘do \(g\) first then \(f\)’
#4F
Inverse functions
Finding the inverse of a function
#4G
Variation, direct and indirect proportion
To include only the following:
\(y ∝ x, y ∝ \dfrac{1}{x}\)
\(y ∝ x^2, y ∝ \dfrac{1}{x^2}\)
\(y ∝ x^3, y ∝ \dfrac{1}{x^3}\)
\(y ∝ \sqrt{x}, y ∝ \dfrac{1}{\sqrt{x}}\)
#4H
Rectangular Cartesian co-ordinates
#4I
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 \((0, c)\)
#4J
Graphs and graphical treatment of the equation:
\(y = Ax^3 + Bx^2 + Cx + D + \dfrac{E}{x} + \dfrac{F}{x^2}\)
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
#4K
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
#4L
Differentiation of integer powers of \(x\)
Use of \(\dfrac{dy}{dx}\) notation
#4M
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
#4N
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:
\(\dfrac{ds}{dt} = v\) and \(\dfrac{dv}{dt} = a\)