FUNCTIONS, INVERSES AND LOGS
Demonstrate an understanding of the definition of a logarithm and any laws needed to solve real-life problems (e.g. growth and decay) (a) Demonstrate the ability to work with various types of functions and relations including the inverses listed in the following Assessment Standard (b) Demonstrate knowledge of the formal definition of a function.
(a) Investigate and generate graphs of the inverse relations of functions, in particular the inverses of:
y = ax + q
y = ax^{2}
y = a^{x}; a > 0
(b) Determine which inverses are functions and how the domain of the original function needs to be restricted so that the inverse is also a function.
Identify characteristics as listed below and hence use applicable characteristics to sketch graphs of the inverses of the functions listed above: (a) domain and range; (b) intercepts with the axes; (c) turning points, minima and maxima; (d) asymptotes; (e) shape and symmetry; (f) average gradient (average rate of change); intervals on which the function increases/decreases.
Clarification
Definition of a logarithm – understand that the logarithmic function is the inverse of the exponential function.
Convert fluently between logarithmic form and exponential form.
The NCS emphasises the use of logarithms to solve practical problems.
Thus
- Complicated logarithm law simplification is not in the spirit of the NCS.
- Solving logarithm equations and inequalities must be seen in the context of functions.
Given the relationship between x and y in:
- a set of graphs
- tables
- words
- algebraic formulae
determine whether the given information represents a function.
Use and interpret functional notation.
In the teaching process learners must understand how f(x) has been transformed to generate:
f(-x)
-f(x)
f(x + a)
f(x) + a
f(ax)
af(x)
x = f(y)