Mathematical functions are the cornerstone of algebra and mathematical analysis. In this lesson, we will highlight functions defined on infinite sets of numbers and piecewise (branched) functions. Additionally, we will explore the concepts of real functions and one-to-one functions, supported by illustrative exercises and their model solutions.
1. Functions on Infinite Sets and Piecewise Functions
What are Functions on Infinite Sets?
These are functions whose domain is an infinite set of numbers, such as functions defined from the set of natural numbers \(\mathbb{N}\) or the set of real numbers \(\mathbb{R}\).
Definition of a Piecewise (Branched) Function
A piecewise function is a function defined by multiple sub-functions (rules), where the specific mathematical rule used changes based on the interval or the set to which the variable \(x\) belongs.
💡 Illustrative Exercise (1): Function of Rational and Irrational Numbers
Let the function \(f\) from \(\mathbb{R}\) to \(\mathbb{R}\) be defined as follows:
- \(f(x) = 5\) if \(x \in \mathbb{Q}\) (\(x\) is a rational number)
- \(f(x) = 0\) if \(x \notin \mathbb{Q}\) (\(x\) is an irrational number)
Required: Find the values of the following images: \(f(2)\), \(f(0.5)\), \(f(\sqrt{3})\), and then determine the domain and range of the function.
Solution:
- Finding \(f(2)\): The number \(2\) is a rational number, therefore: \(f(2) = 5\).
- Finding \(f(0.5)\): The number \(0.5\) is a rational number, therefore: \(f(0.5) = 5\).
- Finding \(f(\sqrt{3})\): The number \(\sqrt{3}\) is an irrational number, therefore: \(f(\sqrt{3}) = 0\).
- Domain (\(D_{f}\)): It is the entire set of real numbers \(\mathbb{R}\).
- Range (\(R_{f}\)): The function only produces two outputs, \(5\) and \(0\). Thus, the range is: \(\{0, 5\}\).
💡 Illustrative Exercise (2): Piecewise Function with Intervals
(Note: The conditions have been mathematically corrected to include the greater-than-or-equal-to sign \(\ge \) in the first rule so that the function is properly defined at zero).
If the function \(f\) from \(\mathbb{R}\) to \(\mathbb{R}\) is defined as follows:
- \(f(x) = x + 1\) if \(x \ge 0\)
- \(f(x) = 2\) if \(x < 0\)
Required: Find the value of each of the following: \(f(3)\), \(f(-5)\), \(f(0)\).
Solution:
To find the required values, we determine the appropriate rule based on the value of \(x\):
To find the required values, we determine the appropriate rule based on the value of \(x\):
- Finding \(f(3)\): Since the number \(3 \ge 0\), we substitute it into the first rule:
\(f(3)=3+1=4\) - Finding \(f(-5)\): Since \(-5 < 0\), we directly use the second rule:
\(f(-5)=2\) - Finding \(f(0)\): Since \(0 \ge 0\) (due to the presence of the greater-than-or-equal-to sign), we substitute it into the first rule:
\(f(0)=0+1=1\)
💡 Illustrative Exercise (3): Describing the Range
A function is defined from \(\mathbb{N}\) to \(\mathbb{N}\) by the rule \(f(x) = 2x\). Describe the range of this function verbally and mathematically.
Solution:
- Verbally: The range is the set of even natural numbers.
- Mathematically: If the set of natural numbers starts from zero, \(\mathbb{N} = \{0, 1, 2, 3, 4, 5, \dots\}\), then the range is:
\(R_{f}=\{0,2,4,6,8,\dots \}\)
2. Real Functions
Definition of a Real Function
A function \(f: A \rightarrow B\) is said to be a real function if both its domain \(A\) and its codomain \(B\) are subsets of the set of real numbers \(\mathbb{R}\). That is:
\(A\subseteq \mathbb{R}\quad \text{and}\quad B\subseteq \mathbb{R}\)
\(A\subseteq \mathbb{R}\quad \text{and}\quad B\subseteq \mathbb{R}\)
💡 Illustrative Exercise (4): Domain of a Radical Function
Let the function be: \(f(x) = \sqrt{x}\)
Required: Find the domain and codomain of the function such that it is a real function.
Solution:
Since the square roots of negative numbers do not yield real numbers, the expression under the radical must be greater than or equal to zero.
Since the square roots of negative numbers do not yield real numbers, the expression under the radical must be greater than or equal to zero.
- Domain (\(A\)): \(A = \{x : x \in \mathbb{R} , x \ge 0\}\) (the set of non-negative real numbers).
- Codomain: It is the set of real numbers \(\mathbb{R}\).
3. One-to-One Function (Injective Function)
Definition of a One-to-One Function
A function \(f: A \rightarrow B\) is said to be a one-to-one function if every element in the codomain \(B\) is the image of at most one element from the domain \(A\).
- Mathematical Formula: A function is one-to-one if and only if the following condition is met:
For all \(x_1, x_2 \in A\), if \(f(x_1) = f(x_2)\), then \(x_1 = x_2\). - Equivalent Formula (Contrapositive):
For all \(x_1, x_2 \in A\), if \(x_1 \neq x_2\), then \(f(x_1) \neq f(x_2)\).
💡 Illustrative Exercise (5): Proving a Function is One-to-One
Let the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) be defined as \(f(x) = x^3\). Prove whether the function is one-to-one or not.
Solution:
- Assume that \(x_1, x_2 \in \mathbb{R}\) such that: \(f(x_1) = f(x_2)\).
- Substituting into the function's rule: \(x_1^3 = x_2^3\).
- Taking the cube root of both sides: \(\sqrt[3]{x_1^3} = \sqrt[3]{x_2^3}\), which leads to: \(x_1 = x_2\).
- Since the assumption \(f(x_1) = f(x_2)\) logically implies that \(x_1 = x_2\), the function is one-to-one.
Frequently Asked Questions (FAQ)
- What is the difference between the domain and the range?
The domain is the set of all possible input values that the variable \(x\) can take, while the range is the set of actual output values produced by the function (\(f(x)\)). - How do I know if a function is piecewise?
A piecewise function is recognized by being written in multiple lines (more than one rule), where each rule is followed by a condition that defines its specific interval.
If you have any questions about real or piecewise functions, leave us a comment below and we will answer you right away!