Applications of Functions

Understanding Functions

A function is a special relation where each input or ‘x-value’ has a single output or ‘y-value’.
A function can be represented in various forms; equations, graphs, or tables.
When writing functions, we commonly use the notation ‘f(x)’ meaning the output is determined by an input ‘x’.
Commonly used functions in mathematics include linear, quadratic, exponential, and trigonometric functions.

Real World Applications of Functions

Functions are used extensively in real-world scenarios and can model phenomena in the fields of physics, economics, population dynamics, medicine, and many others.
The growth or decline of populations, investments, and bacteria can often be modelled with exponential functions.
Linear functions are often applicable in situations where there is a constant rate of change.
Quadratic functions can represent accelerated motion - for instance, the trajectory of a thrown object.

Analysing Functions

When analysing functions, there are key concepts to consider: domain and range, intercepts, intervals of increase or decrease.
The domain of a function is the set of all ‘x-values’ or inputs for which the function is defined.
The range of a function is the set of all possible output or ‘y-values’.
An intercept refers to the point at which the function crosses the x-axis or y-axis.
The function is increasing where the y-value rises as the x-value increases and decreasing where the y-value declines as the x-value increases.

Function Transformations

Functions can be transformed through stretching, reflection, and translation.
Translation is the shifting of a function either vertically (up or down) or horizontally (left or right).
Reflection is when a function is flipped over the x-axis or y-axis.
Stretching or shrinking a function can either be vertically or horizontally.

Function Composition

Composition of functions involves the application of one function to the results of another function.
Notably, the order in which the functions are composed does matter, interestingly f(g(x)) does not necessarily equal g(f(x)).

Remember, tackling this important topic requires not only understanding but also plenty of practice. Make a habit of solving problems from different contexts and increasingly challenging ones.