1.Order pair, Cartesian product and relation

Understanding ordered pairs, Cartesian products, and relations is essential for exploring functions and their graphical representations. Here’s a detailed overview:

1.1 Ordered Pair

An ordered pair is a pair of elements where the order in which the elements are listed matters.

Definition

• An ordered pair is denoted as $(a,b)$, where $a$ is the first element and $b$ is the second element.

Properties

• Order Matters: (a,b)≠(b,a)(a, b) \neq (b, a)(a,b)=(b,a) unless $a=b$.
• Coordinates: In the Cartesian coordinate system, an ordered pair $(x,y)$ represents a point on the plane where $x$ is the x-coordinate and $y$ is the y-coordinate.

Example

• The ordered pair $(3,5)$ represents a point where the x-coordinate is 3 and the y-coordinate is 5.

1.2 Cartesian Product

The Cartesian Product is the set of all possible ordered pairs formed by taking one element from each of two sets.

Definition

• If $A$ and $B$ are sets, the Cartesian product $A\times B$ is defined as: $$A\times B=\{(a,b)\mid a\in A\text{ and }b\in B\}$$

Properties

• Ordered Pairs: Each element of $A\times B$ is an ordered pair where the first component comes from $A$ and the second component comes from $B$.
• Non-Commutative: A×B≠B×AA \times B \neq B \times AA×B=B×A unless $A=B$.

Example

• Let $A=\{1,2\}$ and $B=\{x,y\}$.
  • The Cartesian product $A\times B$ is: $$A\times B=\{(1,x),(1,y),(2,x),(2,y)\}$$

1.3 Relation

A relation between two sets is a collection of ordered pairs where the first element is from one set and the second element is from another set.

Definition

• If $A$ and $B$ are sets, a relation $R$ from $A$ to $B$ is a subset of $A\times B$. Formally: $$R\subseteq A\times B$$

Types of Relations

1. Function: A special type of relation where each element in the domain (first set) is associated with exactly one element in the codomain (second set).
2. One-to-One (Injective): A relation where each element of the domain maps to a unique element of the codomain.
3. Onto (Surjective): A relation where every element of the codomain is associated with at least one element of the domain.
4. One-to-One Correspondence (Bijective): A relation that is both one-to-one and onto.

Example

• Consider sets $A=\{1,2\}$ and $B=\{a,b\}$.
  • A relation $R$ could be: $$R=\{(1,a),(2,b)\}$$
  • This relation pairs elements of $A$ with elements of $B$.

Graph of a Function

The graph of a function is a visual representation of the set of ordered pairs that make up the function. In the Cartesian coordinate system:

• Graph: The graph is the set of points $(x,f(x))$ where $x$ is an element from the domain and $f(x)$ is the corresponding element from the codomain.

Example

• For a function $f$ where f(x)=x2f(x) = x^2f(x)=x2, the graph is the set of points (x,x2)(x, x^2)(x,x2) on the coordinate plane.
• If $f=\{(1,1),(2,4),(3,9)\}$, the graph is a parabola representing the function y=x2y = x^2y=x2.

In mathematics, mapping (or function) is a relation that assigns each element in one set (called the domain) to exactly one element in another set (called the codomain). This concept is crucial in understanding functions and relations.

Definition of Mapping (Function)

A mapping $f:A\to B$ is a rule that assigns each element of set $A$ (the domain) to a unique element in set $B$ (the codomain). For each $a\in A$, there exists a unique $b\in B$ such that $f(a)=b$.

Types of Mapping (Function)

There are three key types of mappings or functions:

1. One-to-One (Injective) Mapping

A function $f:A\to B$ is called one-to-one (or injective) if different elements in the domain $A$ map to different elements in the codomain $B$. In other words, no two distinct elements in $A$ have the same image in $B$.

Formal Definition

• A function $f:A\to B$ is injective if, for all a1,a2∈Aa_1, a_2 \in Aa1​,a2​∈A, f(a1)=f(a2)  ⟹  a1=a2.f(a_1) = f(a_2) \implies a_1 = a_2.f(a1​)=f(a2​)⟹a1​=a2​. This means that if the outputs are the same, the inputs must also be the same.

Example

• Consider the function $f(x)=2x$ from $A=\{1,2,3\}$ to $B=\{2,4,6\}$. The function is one-to-one because: $$f(1)=2,f(2)=4,f(3)=6.$$ Different inputs map to different outputs.

Graphical Representation

In a graph of an injective function, no horizontal line intersects the graph at more than one point.

2. Onto (Surjective) Mapping

A function $f:A\to B$ is called onto (or surjective) if every element in the codomain $B$ is the image of at least one element from the domain $A$. In other words, the function covers the entire codomain.

Formal Definition

• A function $f:A\to B$ is surjective if, for every $b\in B$, there exists at least one $a\in A$ such that: $$f(a)=b.$$ This means that every element of $B$ is “hit” by some element from $A$.

Example

• Consider the function f(x)=x2f(x) = x^2f(x)=x2 from $A=\mathbb{R}$ (real numbers) to $B=[0,\infty )$ (non-negative real numbers). The function is surjective because every non-negative number in $B$ has a corresponding square root in $A$.

Graphical Representation

In a graph of a surjective function, every possible $y$-value in the codomain is “covered” by at least one $x$-value.

3. One-to-One and Onto (Bijective) Mapping

A function $f:A\to B$ is called one-to-one and onto (or bijective) if it is both injective and surjective. This means:

• Every element in the domain maps to a unique element in the codomain (injective).
• Every element in the codomain is mapped by some element from the domain (surjective).

Formal Definition

• A function $f:A\to B$ is bijective if it is both injective and surjective, i.e., ∀a1,a2∈A,f(a1)=f(a2)  ⟹  a1=a2\forall a_1, a_2 \in A, f(a_1) = f(a_2) \implies a_1 = a_2∀a1​,a2​∈A,f(a1​)=f(a2​)⟹a1​=a2​ and for every $b\in B$, there exists an $a\in A$ such that $f(a)=b$.

Example

• Consider the function $f(x)=x+3$ from $A=\mathbb{R}$ to $B=\mathbb{R}$. This function is bijective because:
  • Injective: Each real number $x$ is mapped to a unique real number $x+3$.
  • Surjective: For every real number $y$, there exists a real number $x=y-3$ such that $f(x)=y$.

Graphical Representation

In a graph of a bijective function, each horizontal line intersects the graph at exactly one point. Additionally, each $y$-value in the codomain corresponds to one and only one $x$-value in the domain.

Summary of Mapping Types

Composite and Inverse Functions

In mathematics, composite and inverse functions are essential concepts that help us understand how functions can be combined or “undone.” Let’s explore these concepts in detail.

4.1 Composite Functions

A composite function is the result of applying one function to the result of another function. Given two functions $f$ and $g$, the composite function $f\circ g$ means applying $g$ first and then applying $f$ to the result of $g$.

Definition

• Let $f:B\to C$ and $g:A\to B$ be two functions.
• The composite function $f\circ g$ is defined as: $$(f\circ g)(x)=f(g(x))$$ for all $x\in A$.

Steps to Evaluate a Composite Function

1. Start by evaluating $g(x)$, which gives a result in set $B$.
2. Then, apply $f$ to the result of $g(x)$, giving the final result in set $C$.

Example

Let f(x)=x2f(x) = x^2f(x)=x2 and $g(x)=x+3$.

• To find $(f\circ g)(x)$:

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

• Similarly, to find $(g\circ f)(x)$:

  (g∘f)(x)=g(f(x))=g(x2)=x2+3.(g \circ f)(x) = g(f(x)) = g(x^2) = x^2 + 3.(g∘f)(x)=g(f(x))=g(x2)=x2+3.

Note that, in general, f∘g≠g∘ff \circ g \neq g \circ ff∘g=g∘f.

Properties of Composite Functions

1. Associativity: Composite functions are associative, meaning that: $$f\circ (g\circ h)=(f\circ g)\circ h.$$
2. Identity Function: If $I(x)=x$ is the identity function, then: $$f\circ I=I\circ f=f.$$

4.2 Inverse Functions

An inverse function reverses the effect of a given function. If a function $f$ maps $A$ to $B$, the inverse function f−1f^{-1}f−1 maps $B$ back to $A$.

Definition

• A function $f:A\to B$ is said to have an inverse if there exists a function f−1:B→Af^{-1}: B \to Af−1:B→A such that: f(f−1(x))=xandf−1(f(x))=xf(f^{-1}(x)) = x \quad \text{and} \quad f^{-1}(f(x)) = xf(f−1(x))=xandf−1(f(x))=x for all $x\in A$ and $y\in B$.

Conditions for a Function to Have an Inverse

A function must be bijective (both one-to-one and onto) to have an inverse. This ensures that each element in the domain maps to a unique element in the codomain, and every element in the codomain has a corresponding pre-image in the domain.

Example

• Let $f(x)=2x+3$. To find the inverse function f−1(x)f^{-1}(x)f−1(x):
  1. Start by replacing $f(x)$ with $y$: $$y=2x+3.$$
  2. Solve for $x$ in terms of $y$: x=y−32.x = \frac{y – 3}{2}.x=2y−3​.
  3. Replace $y$ with f−1(x)f^{-1}(x)f−1(x) to get the inverse function: f−1(x)=x−32.f^{-1}(x) = \frac{x – 3}{2}.f−1(x)=2x−3​.

Graphical Interpretation

• The graph of a function and its inverse are reflections of each other across the line $y=x$.
• If a function is invertible, its graph passes the horizontal line test (i.e., no horizontal line intersects the graph more than once).

Composite of a Function and Its Inverse

For a function $f$ and its inverse f−1f^{-1}f−1, the following properties hold:

1. Left Composition: f−1∘f=If^{-1} \circ f = If−1∘f=I (the identity function on the domain of $f$). f−1(f(x))=x.f^{-1}(f(x)) = x.f−1(f(x))=x.
2. Right Composition: f∘f−1=If \circ f^{-1} = If∘f−1=I (the identity function on the codomain of $f$). f(f−1(x))=x.f(f^{-1}(x)) = x.f(f−1(x))=x.

These compositions show that applying a function and then its inverse returns the original input value.

5.Algebraic and transcendental functions

Functions can be classified into two broad categories based on the type of operations they involve: algebraic functions and transcendental functions. These categories help distinguish functions based on whether they can be expressed using basic algebraic operations or require more advanced operations like trigonometric, exponential, or logarithmic functions.

5.1 Algebraic Functions

An algebraic function is a function that can be expressed using a finite number of the basic operations of algebra, such as addition, subtraction, multiplication, division, and the extraction of roots (like square roots).

Definition

• An algebraic function is any function that can be written as the solution to a polynomial equation.
• It can be expressed in the form: $$f(x)=P(x)$$ where $P(x)$ is a polynomial, or it may involve roots or rational expressions.

Types of Algebraic Functions

1. Polynomial Functions: Functions that involve only powers of $x$.
   • Example: f(x)=x3−2x+1f(x) = x^3 – 2x + 1f(x)=x3−2x+1.
2. Rational Functions: Functions that are the ratio of two polynomials.
   • Example: f(x)=2×2+3x−1f(x) = \frac{2x^2 + 3}{x – 1}f(x)=x−12x2+3​.
3. Radical Functions: Functions that involve roots of polynomials.
   • Example: f(x)=x2+1f(x) = \sqrt{x^2 + 1}f(x)=x2+1​.

Examples of Algebraic Functions

1. Linear Function: $f(x)=2x+5$.
2. Quadratic Function: f(x)=x2+4x+4f(x) = x^2 + 4x + 4f(x)=x2+4x+4.
3. Cubic Function: f(x)=x3−3x+1f(x) = x^3 – 3x + 1f(x)=x3−3x+1.
4. Rational Function: f(x)=x2−1x+1f(x) = \frac{x^2 – 1}{x + 1}f(x)=x+1x2−1​.

Properties

• Closed Under Operations: Algebraic functions are closed under addition, subtraction, multiplication, and division.
• Graph Shapes: Polynomial functions tend to have smooth curves, while rational functions often have asymptotes.

5.2 Transcendental Functions

A transcendental function is a function that cannot be expressed in terms of a finite combination of algebraic operations (addition, subtraction, multiplication, division, and root extraction). These functions go beyond algebraic operations and include functions like exponential, logarithmic, and trigonometric functions.

Definition

• Transcendental functions are functions that do not satisfy any algebraic equation involving only polynomials with rational coefficients.

Types of Transcendental Functions

1. Exponential Functions: Functions involving powers where the exponent is a variable.
   • Example: f(x)=exf(x) = e^xf(x)=ex, where $e$ is the base of the natural logarithm.
2. Logarithmic Functions: The inverse of exponential functions.
   • Example: $f(x)=\ln (x)$, where $\ln (x)$ is the natural logarithm.
3. Trigonometric Functions: Functions related to angles and periodic phenomena.
   • Examples: $f(x)=\sin (x),\cos (x),\tan (x)$.
4. Inverse Trigonometric Functions: The inverse of trigonometric functions.
   • Examples: $f(x)=\arcsin (x),\arccos (x),\arctan (x)$.

Examples of Transcendental Functions

1. Exponential Function: f(x)=exf(x) = e^xf(x)=ex.
2. Logarithmic Function: $f(x)=\log (x)$.
3. Sine Function: $f(x)=\sin (x)$.
4. Cosine Function: $f(x)=\cos (x)$.

Properties

• Not Algebraically Expressible: Transcendental functions cannot be written as polynomials or involve only basic algebraic operations.
• Infinite Series Representation: Some transcendental functions can be expressed as infinite series, such as exe^xex being represented by its Taylor series:

  ex=1+x1!+x22!+x33!+⋯e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdotsex=1+1!x​+2!x2​+3!x3​+⋯

• Periodic Nature: Many transcendental functions, such as trigonometric functions, exhibit periodic behavior.

Comparison of Algebraic and Transcendental Functions

6. Functions and Their Graphs

In mathematics, graphs are a powerful visual tool for understanding functions. A graph represents the relationship between the input $x$ and the output $y$ for a function. Let’s explore some key types of functions and their corresponding graphs.

6.1 General Form of Quadratic Equations and Its Graph

A quadratic equation is a second-degree polynomial, which has the general form:

y=ax2+bx+cy = ax^2 + bx + cy=ax2+bx+c

where:

• $a$, $b$, and $c$ are constants.
• a≠0a \neq 0a=0, because if $a=0$, it would be a linear equation.

Graph of a Quadratic Function

The graph of a quadratic function is a parabola, which can either open upwards or downwards depending on the sign of $a$:

• If $a>0$, the parabola opens upward.
• If $a<0$, the parabola opens downward.

Key Features of the Graph

1. Vertex: The highest or lowest point of the parabola. The vertex can be found using the formula:

   xvertex=−b2ax_{\text{vertex}} = -\frac{b}{2a}xvertex​=−2ab​The $y$-coordinate of the vertex is obtained by substituting xvertexx_{\text{vertex}}xvertex​ into the quadratic equation.

2. Axis of Symmetry: The parabola is symmetric about the vertical line passing through the vertex. This line is called the axis of symmetry, given by:

   x=−b2ax = -\frac{b}{2a}x=−2ab​

3. Y-Intercept: The point where the parabola crosses the $y$-axis. It is found by setting $x=0$ in the equation:

   $$y=c$$

4. X-Intercepts (Roots): The points where the parabola crosses the $x$-axis. These can be found by solving the quadratic equation ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0 using the quadratic formula:

   x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}x=2a−b±b2−4ac​​

Example

For the quadratic equation y=x2−4x+3y = x^2 – 4x + 3y=x2−4x+3:

• The vertex is $(2,-1)$.
• The axis of symmetry is $x=2$.
• The graph has two $x$-intercepts at $x=1$ and $x=3$.
• The $y$-intercept is at $y=3$.

The graph is a parabola that opens upwards since $a=1>0$.

6.2 Graph of the Function y=xy = \sqrt{x}y=x​

The function y=xy = \sqrt{x}y=x​ is a square root function, which only takes non-negative values of $x$ since the square root of a negative number is not defined in the real number system.

Graph of y=xy = \sqrt{x}y=x​

• Domain: $[0,\infty )$ (i.e., $x\ge 0$).
• Range: $[0,\infty )$ (i.e., $y\ge 0$).

The graph of y=xy = \sqrt{x}y=x​ is a curve that starts at the origin $(0,0)$ and increases slowly, becoming flatter as $x$ increases. It is a monotonically increasing function, meaning that as $x$ increases, $y$ also increases.

Key Features

1. The graph passes through the point $(0,0)$.
2. It is not defined for negative values of $x$.
3. The graph increases slower as $x$ increases because the square root grows slower than linear functions.

Example

For the function y=xy = \sqrt{x}y=x​, the points $(0,0)$, $(1,1)$, and $(4,2)$ are on the graph. The curve rises quickly at first but then flattens out as $x$ increases.

6.3 System of Homogeneous Equations and Their Graph

A system of homogeneous equations consists of two or more equations where each equation is set to zero. A typical system of homogeneous linear equations looks like:

a1x+b1y=0a_1x + b_1y = 0a1​x+b1​y=0 a2x+b2y=0a_2x + b_2y = 0a2​x+b2​y=0

Graph of Homogeneous Equations

The graph of each linear equation in the system is a straight line that passes through the origin $(0,0)$, because for homogeneous systems, the constant term is zero. The solution to the system of homogeneous equations is the set of points where the lines intersect. For most cases, the only solution is the trivial solution at the origin $(0,0)$.

Key Scenarios for Two Equations

1. Coincident Lines: If the two equations represent the same line (i.e., they are scalar multiples of each other), the solution is all points on that line.
2. Distinct Lines: If the lines are distinct and not multiples of each other, they intersect at exactly one point—the origin $(0,0)$.
3. Parallel Lines: If the lines are parallel but not the same, they do not intersect, and the only solution is the origin.

Example

Consider the system:

$$2x+3y=0$$ $$4x+6y=0$$

These two equations represent the same line because the second equation is a multiple of the first. Therefore, the solution is all points on the line $2x+3y=0$.

Another example:

$$x+y=0$$ $$2x-y=0$$

In this case, the lines intersect only at the origin $(0,0)$, so the solution is $(0,0)$.