Summary (Grade 10 NSC Matric Mathematics): Revision Notes

Summary

Functions are mathematical relationships that connect input values (x) to output values (y) according to specific rules. Understanding functions is essential for solving many mathematical problems and interpreting real-world relationships.

Key characteristics of functions

Independent and dependent variables form the foundation of any function. The independent variable (usually x) represents the input value that you can choose freely. The dependent variable (usually y) represents the output value that depends entirely on what you choose for the independent variable.

Domain and range define the boundaries of a function. The domain includes all possible x-values that you can use as inputs for a particular function. For each x-value in the domain, there should be at most one corresponding y-value. The range contains all possible y-values that the function can produce when you use at least one x-value from the domain.

Asymptotes are imaginary straight lines that a function's graph approaches but never actually touches or crosses. These lines help us understand the behaviour of functions, especially when dealing with values that approach infinity or create undefined situations.

Types of functions and their forms

Mathematics recognises several important categories of functions, each with distinctive characteristics and applications:

Linear functions follow the form $y = axe + q$, where 'a' represents the gradient (slope) and 'q' represents the y-intercept. These functions create straight-line graphs and show constant rates of change.

Parabolic functions use the form $y = ax^2 + q$, where the squared term creates the characteristic U-shaped or inverted U-shaped curves called parabolas. The value of 'a' determines whether the parabola opens upward or downward.

Hyperbolic functions take the form $y = \frac{a}{x} + q$, creating curved graphs with asymptotes. These functions are undefined when x equals zero, which creates interesting behaviour patterns.

Exponential functions follow $y = ab^x + q$, where the variable appears as an exponent. These functions show rapid growth or decay and are crucial for modelling population changes, radioactive decay, and compound interest.

Trigonometric functions include three main forms: $y = a \sin θ + q$, $y = a \cos θ + q$, and $y = a \tan θ + q$. These functions repeat in predictable patterns and are essential for describing periodic phenomena like waves and oscillations.

Function table examples

Function tables help visualise the relationship between input and output values. Here are some examples:

This table shows a pattern where y-values increase consistently, suggesting a linear relationship.

This table demonstrates how functions can produce both positive and negative outputs depending on the input values.

This simple table represents the function $y = x$, where every input equals its corresponding output.

Working with function problems

When identifying functions, remember that each x-value must correspond to exactly one y-value.

When determining which function passes through a specific point like (1, 7), substitute the coordinates into each option and check which equation holds true.