Functions are at the heart of algebra, providing a systematic way to express the relationship between variables. This section introduces the concept of functions, their domain and range, and how to use function notation, focusing on mathematical precision and clarity.
![Function illustration](https://tutorchase-resources-platform-client-production.vercel.app/_next/image?url=https%3A%2F%2Ftutorchase-production.s3.eu-west-2.amazonaws.com%2F54ea1b35-7330-429e-acb0-03184ccb5469-file.png&w=1080&q=75)
Image courtesy of Course Sidekick
What is a Function?
A function defines a specific relationship where each input value (x) corresponds to exactly one output value, denoted as (f(x)). Functions can be represented in various forms: as equations, graphs, or tables.
Domain and Range
Domain
The domain of a function is the set of all possible input values for which the function is defined.
Range
The range of a function includes all possible output values that result from using the domain.
![Domain and Range](https://tutorchase-resources-platform-client-production.vercel.app/_next/image?url=https%3A%2F%2Ftutorchase-production.s3.eu-west-2.amazonaws.com%2F9a426e6b-9b6f-4c7e-9244-1c0806af46f2-file.png&w=1080&q=75)
Image courtesy of Cuemath
The domain and range of this function is given as domain , range .
Function Notation
Function notation, such as , provides a convenient way to reference functions and perform operations with them.
Worked Examples
Example 1: Linear Function
- Domain: All real numbers .
- Range: All real numbers .
Example 2: Rational Function
- Domain: All real numbers .
- Range: All real numbers .
Practical Problems
Problem 1:
Find the output for when .
Given:
Solution:
To find :
Problem 2:
Find the output for when .
Given:
Solution:
To find :