Functions and Relation

Functions and Relations

Lesson Objectives

Define relations and functions and understand their differences.
Identify domain, codomain, and range of functions.
Determine if a relation is a function using the vertical line test.
Understand and work with types of functions: one-to-one, onto, and inverse functions.
Perform function operations and composition of functions.

Lesson Introduction

In mathematics, relations link elements from one set to another. A function is a special type of relation where each input corresponds to exactly one output. Understanding functions is key to modeling real-world phenomena and solving complex problems.

Lesson Content

Relations and Functions

A relation from set $A$ to set $B$ is a subset of the Cartesian product $A \times B$ . A function $f$ from $A$ to $B$ is a relation where every element in $A$ has exactly one image in $B$ .

Example 1: Let

A = \{1, 2, 3\}

B = \{4, 5, 6\}

. Which of the following are functions?

$\{(1,4), (2,5), (3,6)\}$ — Yes, because each input maps to exactly one output.
$\{(1,4), (2,5), (2,6)\}$ — No, because input 2 maps to two outputs (5 and 6).

Domain, Codomain, and Range

- The domain is the set of all inputs.
- The codomain is the set of potential outputs.
- The range is the set of actual outputs from the function.

Example 2: For

f: \mathbb{R} \to \mathbb{R}, \quad f(x) = x^2

Domain: $\mathbb{R}$
Codomain: $\mathbb{R}$
Range: $\{y \in \mathbb{R} \mid y \geq 0\}$

Vertical Line Test

A relation graphed in the coordinate plane is a function if and only if no vertical line intersects the graph more than once.

Types of Functions

- One-to-one (Injective): No two inputs have the same output.
- Onto (Surjective): Every element in the codomain is an output of the function.
- Bijective: Both one-to-one and onto.

Example 3: Determine if

f(x) = 2x + 3

is one-to-one and onto when

f: \mathbb{R} \to \mathbb{R}

.
It is one-to-one since different

x

values give different outputs, and onto because for any

y \in \mathbb{R}

, there exists

x = \frac{y - 3}{2}

such that

f(x) = y

Inverse Functions

If a function $f$ is bijective, then it has an inverse $f^{-1}$ such that $f^{-1}(f(x)) = x$ .

Example 4: Find the inverse of $f(x) = 3x - 7$ . Swap $x$ and $y$ : $x = 3y - 7$ . Solving for $y$ : $y = \frac{x + 7}{3}$ . So, $f^{-1}(x) = \frac{x + 7}{3}$ .

Function Operations

For two functions $f$ and $g$ , we can define:
- Sum: $(f+g)(x) = f(x) + g(x)$
- Difference: $(f-g)(x) = f(x) - g(x)$
- Product: $(fg)(x) = f(x) \cdot g(x)$
- Quotient: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}, \quad g(x) \neq 0$

Example 5: If

f(x) = x^2

and

g(x) = x + 1

, find

(f+g)(x)

and

(fg)(x)

(f+g)(x) = x^2 + x + 1

(fg)(x) = x^2(x + 1) = x^3 + x^2

Composition of Functions

The composition $f \circ g$ means applying $g$ first, then $f$ . Formally:
$(f \circ g)(x) = f(g(x))$

Example 6: Given

f(x) = 2x + 3

g(x) = x^2

, find

(f \circ g)(x)

and

(g \circ f)(x)

(f \circ g)(x) = f(g(x)) = f(x^2) = 2x^2 + 3

(g \circ f)(x) = g(f(x)) = (2x + 3)^2 = 4x^2 + 12x + 9

Exercises

Identify whether the relation $\{(2,5), (3,7), (4,5), (2,6)\}$ is a function.
[WAEC]Find the domain and range of $f(x) = \sqrt{x-1}$ .[past-question]
Determine if $f(x) = x^3$ is one-to-one and onto from $\mathbb{R} \to \mathbb{R}$ .
[NECO]Find the inverse of $f(x) = \frac{x-4}{5}$ .[past-question]
Given $f(x) = x^2$ , $g(x) = 3x + 1$ , find $(f-g)(x)$ .
Calculate $(g \circ f)(2)$ if $f(x) = x + 1$ and $g(x) = 2x^2$ .
[WAEC]Show that $f(x) = x^2$ is not one-to-one by giving two distinct inputs with the same output.[past-question]
If $f(x) = \frac{1}{x}$ , evaluate $f(f(2))$ .
[NECO]Given $f(x) = x^2 - 4x + 3$ , find all values of $x$ for which $f(x) = 0$ .[past-question]
Sketch the graph of $y = |x - 2|$ for $-2 \leq x \leq 6$ .

Conclusion/Recap

A relation is a function if every input has exactly one output. The domain of a function is the set of all possible input values, while the range is the set of possible outputs. One-to-one functions map distinct inputs to distinct outputs. Onto functions cover the entire codomain with their outputs. Function operations include addition, subtraction, multiplication, and composition. The inverse of a function “undoes” the action of the function, when it exists. Graphs and tables can be used to study the behavior and properties of functions.