Summary (Grade 12 NSC Matric Mathematics): Revision Notes

Summary

What is a function?

A function is a special type of rule that creates a unique relationship between two sets of values. Think of it as a mathematical machine - for every input you put in, you get exactly one output.

More formally, a function uniquely associates each element from one set (called set A) with exactly one element from another set (called set B).

Functions can be classified into different types based on how they map inputs to outputs. A one-to-one relation means each input corresponds to a unique output, whilst a many-to-one relation allows multiple inputs to produce the same output. However, for something to be a function, each input can only produce one output.

Testing if a relation is a function

Understanding inverse functions

An inverse function, written as $f^{-1}(x)$, essentially "undoes" what the original function does. If $f(x)$ takes you from input to output, then $f^{-1}(x)$ takes you back from output to input.

Domain and range relationships

Special types of functions and their inverses

Straight line functions

Straight line functions have the form $y = axe + q$, where $a$ and $q$ are constants. Their inverse functions are also straight lines (with the important exception of vertical and horizontal lines, which don't have inverses that are functions).

The graph above shows a straight line with x-intercept at (-3, 0) and y-intercept at (0, -6), demonstrating the linear relationship between x and y coordinates.

Quadratic functions

Quadratic functions follow the pattern $y = ax^2$. Interestingly, the inverse of a parabola is not actually a function because it fails the vertical line test. However, you can make it work by limiting the domain of the original parabola so that its inverse becomes a function.

Exponential functions

Exponential functions have the form $y = b^x$ where $b > 0$ and $b \neq 1$. The inverse of an exponential function is a logarithmic function: $f^{-1}(x) = \log_b x$.

Logarithmic functions and laws

The logarithmic function is closely related to exponential functions. The common logarithm uses base 10 and can be written as $\log_{10} x = \log x$. When you see the log symbol without a base specified, it means base 10.

Comparing function types

Function Type	Formula	Inverse Formula	Has Inverse Function?
Straight line	$y = axe + q$	$y = \frac{x - q}{a}$	Yes
Quadratic	$y = ax^2$	$y = \pm\sqrt{\frac{x}{a}}$	No (unless domain restricted)
Exponential	$y = b^x$	$y = \log_b x$	Yes

Understanding these patterns helps you quickly identify function types and predict their behaviour, which is especially useful in exam situations.