Topics

Domain and Range

 Domain and Range: A Comprehensive Guide

 

 General Concepts

 

 Domain

- The set of all possible input values (x-values) for which the function is defined

- Denoted as: D(f) or Dom(f)

 

 Range

- The set of all possible output values (y-values) that result from inputting domain values

- Denoted as: R(f) or Ran(f)

 

 General Method to Find Domain

 

1. Start with ℝ (all real numbers)

2. Eliminate values that make the function undefined:

   - Denominators cannot equal zero

   - Even roots cannot have negative radicands

   - Logarithms must have positive arguments

   - Domain restrictions specific to given context

 

 Example Process:

1. Write out the function

2. Check for:

   - Division by zero

   - Square roots or even roots

   - Logarithms

   - Contextual restrictions

3. Combine all restrictions

4. Write in interval notation

 

 General Method to Find Range

 

1. Algebraic Method:

   - Solve x in terms of y

   - Find restrictions on y-values

   - Consider function behavior

  

2. Graphical Method:

   - Sketch the graph

   - Find lowest/highest points

   - Check for asymptotes

   - Look for gaps

 

 Function-Specific Methods

 

 1. Linear Functions

f(x) = mx + b

 

Domain:

- Always all real numbers: (-∞, ∞)

- No restrictions unless given in context

 

Range:

- Always all real numbers: (-∞, ∞)

- Method: Since m≠0, function can achieve any y-value

 

 2. Quadratic Functions

f(x) = ax² + bx + c

 

Domain:

- Always all real numbers: (-∞, ∞)

 

Range:

Method:

1. Convert to vertex form: a(x-h)² + k

2. If a > 0: Range is [k, ∞)

3. If a < 0: Range is (-∞, k]

where k is the y-coordinate of vertex

 

 3. Rational Functions

f(x) = P(x)/Q(x)

 

Domain:

Method:

1. Set Q(x) ≠ 0

2. Solve for excluded x-values

3. Write domain as ℝ minus these values

 

Range:

Method:

1. Find horizontal asymptotes

2. Find critical points

3. Consider behavior between asymptotes

 

 4. Root/Radical Functions

f(x) = ⁿ√x

 

Domain:

Method:

1. If n is even: x ≥ 0

2. If n is odd: all real numbers

 

Range:

Method:

1. If n is even: [0, ∞)

2. If n is odd: (-∞, ∞)

 

 5. Exponential Functions

f(x) = aˣ (a > 0, a ≠ 1)

 

Domain:

- Always all real numbers: (-∞, ∞)

 

Range:

- Always (0, ∞)

- Never includes 0 or negative numbers

 

 6. Logarithmic Functions

f(x) = logₐ(x)

 

Domain:

- Always (0, ∞)

- Method: Set argument > 0

 

Range:

- Always (-∞, ∞)

- Can achieve any real number

 

 7. Trigonometric Functions

 

Sine:

- Domain: (-∞, ∞)

- Range: [-1, 1]

 

Cosine:

- Domain: (-∞, ∞)

- Range: [-1, 1]

 

Tangent:

- Domain: x ≠ π/2 + πn, n ∈ ℤ

- Range: (-∞, ∞)

 

 How Function Properties Affect Domain and Range

 

 1. Composition

For f∘g(x):

- Domain: Values in domain of g where g(x) is in domain of f

- Range: Range of f restricted to inputs from g

 

 2. Transformations

 

Vertical Shifts [f(x) ± k]:

- Domain: Unchanged

- Range: Shifted up/down by k

 

Horizontal Shifts [f(x ± k)]:

- Domain: Shifted left/right by ∓k

- Range: Unchanged

 

Vertical Stretches [kf(x)]:

- Domain: Unchanged

- Range: Multiplied by k

 

Reflections:

- Over x-axis [-f(x)]: Domain unchanged, Range negated

- Over y-axis [f(-x)]: Domain negated, Range unchanged

 

 3. Piecewise Functions

Method:

1. Find domain/range for each piece

2. Domain: Union of individual domains

3. Range: Union of individual ranges

 

 Common Restrictions on Domain

 

1. Division:

   - Cannot divide by zero

   - Example: 1/x, domain: x ≠ 0

 

2. Even Roots:

   - Radicand must be non-negative

   - Example: √x, domain: x ≥ 0

 

3. Logarithms:

   - Argument must be positive

   - Example: ln(x), domain: x > 0

 

4. Inverse Trigonometric:

   - arcsin(x): [-1, 1]

   - arccos(x): [-1, 1]

   - arctan(x): (-∞, ∞)

 

 Tips for Finding Range

 

1. Look for Asymptotes:

   - Horizontal asymptotes limit range

   - Vertical asymptotes affect domain

 

2. Find Critical Points:

   - Maximum and minimum values

   - Points where derivative is zero or undefined

 

3. Consider Function Type:

   - Polynomial degree affects range

   - Rational functions may have restrictions

   - Periodic functions have repeating ranges

 

4. Check Endpoints:

   - For restricted domains

   - For piecewise functions

 

 Common Mistakes to Avoid

 

1. Forgetting to check denominator = 0

2. Ignoring domain restrictions when finding range

3. Not considering holes in rational functions

4. Assuming transformations affect domain/range uniformly

5. Forgetting context-based restrictions

Algebra of Functions, Graph of Functions

Function Graphs and Analysis: A Comprehensive Guide

 

 Basic Principles of Graphing

 

 Key Points to Plot

1. Intercepts

   - x-intercepts: f(x) = 0

   - y-intercept: x = 0

   - Represent roots and initial values

 

2. Critical Points

   - Where f'(x) = 0 or undefined

   - Local maxima and minima

   - Points of inflection

 

3. Asymptotes

   - Horizontal: lim(x→±∞) f(x)

   - Vertical: where denominator = 0

   - Slant: for rational functions

 

 Graph Sketching Steps

1. Find domain and range

2. Identify intercepts

3. Find critical points

4. Determine asymptotes

5. Test intervals for behavior

6. Plot key points

7. Connect points considering continuity

 

 Analyzing Roots Through Graphs

 

 Properties of Roots

- x-intercepts = roots

- y-value = 0 at roots

- Crossing x-axis = odd multiplicity

- Touching x-axis = even multiplicity

 

 Number of Roots

1. Polynomial Degree

   - Maximum roots = degree

   - Complex roots come in pairs

 

2. Intermediate Value Theorem

   - If f(a)·f(b) < 0, root exists in [a,b]

   - Helps locate real roots

 

3. Visual Identification

   - Count x-axis crossings

   - Include tangent points

   - Consider multiplicity

 

 Graphing Different Function Types

 

 1. Linear Functions (f(x) = mx + b)

1. Plot y-intercept (0,b)

2. Use slope to plot second point

3. Draw straight line

 

 2. Quadratic Functions (f(x) = ax² + bx + c)

1. Find vertex: x = -b/(2a)

2. Find y-intercept: f(0)

3. Find x-intercepts if any

4. Plot parabola through points

5. Opens up if a > 0, down if a < 0

 

 3. Polynomial Functions

1. Find all intercepts

2. Find critical points

3. Determine end behavior

4. Plot key points

5. Connect smoothly

 

 4. Rational Functions

1. Find vertical asymptotes

2. Find horizontal asymptote

3. Find x,y-intercepts

4. Identify holes

5. Plot critical points

6. Connect considering asymptotes

 

 5. Exponential Functions

1. Plot y-intercept (0,1)

2. Use key points (1,a), (-1,1/a)

3. Draw smooth curve

4. Note horizontal asymptote

 

 6. Logarithmic Functions

1. Plot x-intercept (1,0)

2. Use key points

3. Draw smooth curve

4. Note vertical asymptote

 

 7. Trigonometric Functions

1. Mark period on x-axis

2. Plot key angles (0,π/6,π/4,π/3,π/2...)

3. Connect points smoothly

4. Note amplitude and period

 

 Graphing Composite Functions

 

 General Method

1. Work inside out

2. Track domain restrictions

3. Map points step by step

4. Consider effect of each function

 

 Common Compositions

 

1. f(g(x))

   - First graph y = g(x)

   - Then map points through f

   - Check domain restrictions

 

2. f(x) ± g(x)

   - Graph both functions

   - Add/subtract y-values

   - Note shared domain

 

3. f(x)·g(x)

   - Multiply y-values

   - Check zero product

   - Note domain intersection

 

 Trigonometric Compositions

 

1. Inside Compositions: f(trigx)

Example: ln(sinx)

- Find domain: sinx > 0

- Mark period of sinx

- Apply ln to y-values

- Note undefined regions

 

2. Outside Compositions: trig(f(x))

Example: sin(x²)

- Graph x²

- Apply sine to y-values

- Note amplitude remains [-1,1]

- Period varies with x

 

3. Multiple Compositions

Example: sin(x²)cos(x)

1. Graph x²

2. Graph cos(x)

3. Multiply results

4. Note varying amplitude

 

 Special Considerations for Trig Compositions

 

1. Period Changes

- Inside function affects period

- Check at multiples of π

- Consider phase shifts

 

2. Amplitude Changes

- Outside function affects amplitude

- Check maximum/minimum

- Consider vertical stretches

 

3. Domain Restrictions

- Check each component

- Consider intersection

- Note undefined points

 

 Graph Transformations

 

 Basic Transformations

1. Vertical Shifts: f(x) ± k

   - Up/down k units

 

2. Horizontal Shifts: f(x ± k)

   - Left/right k units

 

3. Stretches: af(x)

   - Vertical stretch by |a|

   - Reflection if a < 0

 

4. Horizontal Stretches: f(ax)

   - Compress if |a| > 1

   - Stretch if |a| < 1

 

 Combining Transformations

1. Work inside out

2. Apply shifts last

3. Order matters

4. Check key points

 

 Analyzing Function Behavior

 

 Through Graphs

1. Increasing/Decreasing

   - Slope positive/negative

   - Read left to right

 

2. Concavity

   - Opens up/down

   - Inflection points

 

3. Continuity

   - No breaks/jumps

   - Check removable discontinuities

 

4. Extrema

   - Peaks and valleys

   - Global vs local

 

 Common Mistakes to Avoid

 

1. Forgetting domain restrictions

2. Incorrect asymptote behavior

3. Wrong transformation order

4. Missing critical points

5. Incorrect root multiplicity

6. Ignoring end behavior

 

 Tips for Complex Graphs

 

1. Use technology for verification

2. Sketch basic shape first

3. Add details systematically

4. Check special points

5. Verify with derivatives

6. Consider symmetry

 

See example of some graphs of composite functions:

Checout how these graphs are drawn:

 Detailed Steps for Graphing Composite Functions

 

 1. f(x) = sin(x²)

 

 Step-by-Step Process

1. Analyze Inner Function (x²)

   - Parabola opening upward

   - Always non-negative

   - Increasing for x > 0, decreasing for x < 0

 

2. Apply Outer Function (sin)

   - Domain: All real numbers (both functions defined everywhere)

   - Range: [-1, 1] (sine's range)

   - Period: Not constant (varies with x)

  

3. Key Points Analysis

   - At x = 0: x² = 0, so sin(0) = 0

   - When x² = π/2: sin(x²) = 1

   - When x² = π: sin(x²) = 0

   - When x² = 3π/2: sin(x²) = -1

 

4. Behavior Analysis

   - As |x| increases, x² increases faster

   - Oscillations become more frequent

   - Amplitude remains ±1

 

5. Graphing Process

  

   a. Plot origin point (0,0)

   b. Find x values where x² = π/2, π, 3π/2, 2π

   c. Mark corresponding y-values (1, 0, -1, 0)

   d. Connect points with increasingly rapid oscillations

   e. Graph is symmetric about y-axis (even function)

  

 

 2. f(x) = sin(x)cos(x)

 

 Step-by-Step Process

1. Identify Alternative Form

   - sin(x)cos(x) = ½ sin(2x)

   - This identity helps understand the behavior

 

2. Analyze Components

  

   sin(x): period 2π, range [-1,1]

   cos(x): period 2π, range [-1,1]

   Product: period π, range [-½,½]

  

 

3. Key Points

   - At x = 0: sin(0)cos(0) = 0

   - At x = π/4: sin(π/4)cos(π/4) = ½

   - At x = 3π/4: sin(3π/4)cos(3π/4) = -½

   - At x = π: sin(π)cos(π) = 0

 

4. Graphing Process

  

   a. Plot zeros at x = 0, π/2, π, 3π/2, 2π

   b. Plot maxima (½) at x = π/4 + 2πn

   c. Plot minima (-½) at x = 3π/4 + 2πn

   d. Connect points with smooth curve

   e. Note: Graph repeats every π

  

 

 3. f(x) = ln(sin(x))

 

 Step-by-Step Process

1. Domain Analysis

   - Inner function (sin x) range: [-1,1]

   - ln(x) requires x > 0

   - Therefore, domain: x where sin(x) > 0

   - Valid in intervals (2πn, π + 2πn)

 

2. Key Points

  

   At x = π/2 + 2πn: sin(x) = 1, so ln(1) = 0

   As x → 0⁺: ln(sin x) → -∞

   As x → π⁻: ln(sin x) → -∞

  

 

3. Behavior Analysis

   - Vertical asymptotes at x = 0, π, 2π, ...

   - Maximum points at x = π/2 + 2πn

   - Function undefined when sin(x) ≤ 0

 

4. Graphing Process

  

   a. Mark vertical asymptotes

   b. Plot maximum points (0,0)

   c. Note rapid decrease near asymptotes

   d. Connect with smooth curves

   e. Show undefined regions with dashed lines

  

 

 4. f(x) = sin(1/x)

 

 Step-by-Step Process

1. Domain Analysis

   - 1/x undefined at x = 0

   - Domain: all real numbers except 0

 

2. Behavior Near Zero

   - As x → 0⁺, 1/x → ∞

   - As x → 0⁻, 1/x → -∞

   - Oscillations become infinitely rapid

 

3. Key Points Analysis

  

   For large |x|: 1/x → 0, so sin(1/x) → 0

   When 1/x = π/2: x = 2/π, y = 1

   When 1/x = 3π/2: x = 2/3π, y = -1

  

 

4. Behavior Analysis

   - Oscillations slow down as |x| increases

   - Function approaches 0 as |x| → ∞

   - No period, but bounded by y = ±1

 

5. Graphing Process

  

   a. Mark vertical asymptote at x = 0

   b. Plot points where sin(1/x) = ±1

   c. Show rapid oscillations near x = 0

   d. Show dampening oscillations for large |x|

   e. Keep curve within bounds y = ±1

  

 

 General Tips for Graphing Composite Functions

 

1. Inside-Out Analysis

   - Start with innermost function

   - Track how each function modifies the input

   - Consider domain restrictions at each step

 

2. Key Features to Track

  

   - Domain changes

   - Range restrictions

   - Periodicity changes

   - Asymptotic behavior

   - Symmetry properties

  

 

3. Useful Techniques

   - Use test points

   - Consider end behavior

   - Check for symmetry

   - Identify critical points

   - Note discontinuities

 

4. Common Patterns

  

   Composition with x²: Even function result

   Composition with 1/x: Often creates oscillations

   Composition with ln or exp: Changes amplitude behavior

   Multiple trig functions: Consider identities

  

 

5. Verification Steps

   - Check domain restrictions

   - Verify key points

   - Confirm asymptotic behavior

   - Test function values

   - Consider symmetry properties

Algebra of Functions, Types of Relations (Reflexive, Symmetric, Transitive, Equivalence)

 Algebra of Functions and Types of Relations

 

 Part 1: Algebra of Functions

 

 1. Basic Operations on Functions

 

 A. Addition of Functions (f + g)

- Definition: (f + g)(x) = f(x) + g(x)

- Domain: Intersection of domains of f and g

- Example: If f(x) = x² and g(x) = x, then (f + g)(x) = x² + x

 

 B. Subtraction of Functions (f - g)

- Definition: (f - g)(x) = f(x) - g(x)

- Domain: Intersection of domains of f and g

- Example: If f(x) = x² and g(x) = x, then (f - g)(x) = x² - x

 

 C. Multiplication of Functions (f·g)

- Definition: (f·g)(x) = f(x)·g(x)

- Domain: Intersection of domains of f and g

- Example: If f(x) = x and g(x) = x + 1, then (f·g)(x) = x(x + 1)

 

 D. Division of Functions (f/g)

- Definition: (f/g)(x) = f(x)/g(x)

- Domain: Intersection of domains of f and g, excluding where g(x) = 0

- Example: If f(x) = x² and g(x) = x, then (f/g)(x) = x where x ≠ 0

 

 2. Composition of Functions

 

 Definition and Notation

- (f∘g)(x) = f(g(x))

- Read as "f composed with g"

- Domain: x values where g(x) is in domain of f

 

 Properties

1. Not Commutative

   - Generally, f∘g ≠ g∘f

 

2. Associative

   - (f∘g)∘h = f∘(g∘h)

 

3. Identity Function

   - i∘f = f∘i = f

   - Where i(x) = x

 

 3. Special Properties

 

 A. Identity Element

- Under addition: Zero function f(x) = 0

- Under multiplication: Unit function f(x) = 1

 

 B. Inverse Elements

- Additive inverse: -f(x)

- Multiplicative inverse: 1/f(x), when f(x) ≠ 0

 

 C. Distributive Property

- f∘(g + h) = f∘g + f∘h (if f is linear)

 

 Part 2: Relations

 

 1. Basic Concepts

 

 Definition

- A relation R from set A to set B is a subset of A × B

- Notation: aRb means (a,b) ∈ R

 

 Ways to Represent Relations

1. Set of Ordered Pairs

   - R = {(a,b) | a ∈ A, b ∈ B}

 

2. Arrow Diagram

   - Visual representation showing connections

 

3. Matrix Form

   - Using 1's and 0's to show relationships

 

4. Graph

   - For numeric relations

 

 2. Types of Relations

 

 A. Reflexive Relation

- Definition: aRa for all a ∈ A

- Matrix: Principal diagonal all 1's

- Example: "Equal to" relation

- Test: Check if (a,a) ∈ R for all a

 

 B. Symmetric Relation

- Definition: If aRb then bRa

- Matrix: Symmetric about principal diagonal

- Example: "Is sibling of" relation

- Test: If (a,b) ∈ R, check if (b,a) ∈ R

 

 C. Transitive Relation

- Definition: If aRb and bRc then aRc

- Example: "Greater than" relation

- Test: For any path a→b→c, check if a→c exists

 

 D. Anti-symmetric Relation

- Definition: If aRb and bRa then a = b

- Example: "Less than or equal to" relation

- Test: Check if mutual relations only exist for same elements

 

 3. Equivalence Relations

 

 Definition

A relation R is an equivalence relation if it is:

1. Reflexive

2. Symmetric

3. Transitive

 

 Properties

1. Partitions set into equivalence classes

2. Each element belongs to exactly one class

3. Classes are disjoint

 

 Examples

1. "Has same remainder when divided by n"

2. "Is parallel to" (for lines)

3. "Has same birthday as"

 

 4. Testing Relations

 

 Step-by-Step Process

1. For Reflexivity

   ```

   Check if (a,a) exists for all a

   Look for all diagonal entries in matrix

   ```

 

2. For Symmetry

   ```

   For each (a,b), check if (b,a) exists

   Compare entries across diagonal in matrix

   ```

 

3. For Transitivity

   ```

   For each pair of relations (a,b) and (b,c)

   Check if (a,c) exists

   ```

 

 5. Equivalence Classes

 

 Definition

- [a] = {x ∈ A | xRa}

- All elements related to a

 

 Properties

1. Union of all classes = original set

2. Classes are disjoint

3. Each class contains related elements

 

 6. Partial Order Relations

 

 Definition

A relation R is a partial order if it is:

1. Reflexive

2. Anti-symmetric

3. Transitive

 

 Examples

1. ≤ on real numbers

2. ⊆ on sets

3. | (divides) on integers

 

 7. Applications

 

 A. Set Theory

- Subset relation

- Equality relation

- Element-of relation

 

 B. Number Theory

- Congruence relations

- Divisibility relations

- Order relations

 

 C. Computer Science

- Database relations

- Graph relationships

- Equality testing

 

 D. Real-World Examples

1. Family relationships

2. Social network connections

3. Organizational hierarchies

Cartesian Product of Sets

Cartesian Product of Sets: Advanced Concepts and Application Examples

 

Conceptual Overview:

The Cartesian product of two sets, \( A \) and \( B \), denoted as \( A \times B \), is a set of ordered pairs where the first element is from \( A \) and the second from \( B \). This concept is foundational in creating relations and functions within advanced mathematics.

 

\[A \times B = \{(a, b) | a \in A, b \in B\}\]

 

Example Applications in JEE Advanced:

1. Relations and Functions: Define a function \( f: A \rightarrow B \) where the graph of \( f \) is a subset of \( A \times B \). Understanding and visualizing these as sets of ordered pairs can help in determining function behavior or finding inverse functions.

 

2. Probability and Combinatorics: Problems might involve selecting items from two different categories, modeled by \( A \times B \), to study various outcome combinations and their probabilities.

 

3. Geometry and Coordinate Systems: The Cartesian product is essential for defining points in planes, spaces, or higher dimensions, where coordinates are represented as tuples derived from Cartesian products of sets, typically \( \mathbb{R} \times \mathbb{R} \) for the Euclidean plane.

 

 Advanced Questions Using Cartesian Products for Practice:

Certainly! Let’s revise and complete the practice questions, ensuring each one includes a full solution and the methods for solving them, particularly within the context of the Cartesian Product of Sets. This content will be suitable for JEE Advanced preparation and include relevant advanced concepts.

 

 Detailed Solutions to Practice Questions on Cartesian Product of Sets:

 

Question 1: Cartesian Products and Set Order

Given two sets \( A = \{1, 2, 3\} \) and \( B = \{x, y\} \), calculate \( A \times B \) and \( B \times A \), and explain why these products are not equal.

 

Solution:

- Calculate \( A \times B \):

  \[A \times B = \{(1, x), (1, y), (2, x), (2, y), (3, x), (3, y)\}\]

- Calculate \( B \times A \):

  \[B \times A = \{(x, 1), (x, 2), (x, 3), (y, 1), (y, 2), (y, 3)\}\]

- Explanation: The sets \( A \times B \) and \( B \times A \) contain ordered pairs where the ordering of elements defines different relationships. The first set takes elements from \( A \) first and \( B \) second, while the second set reverses this order. Since the order in pairs matters, \( A \times B \neq B \times A \).

 

Question 2: Relations Defined by Cartesian Products

Consider a set \( A \) of numbers 1 through 5, and set \( B \) of letters a through e. Define a relation \( R \) on \( A \times B \) such that \( (a, b) \in R \) if and only if the position of \( b \) in the alphabet is less than or equal to \( a \). List the elements of \( R \).

 

Solution:

- Set Definitions: \( A = \{1, 2, 3, 4, 5\} \), \( B = \{a, b, c, d, e\} \)

- Relation \( R \):

  \[R = \{(1, a), (2, a), (2, b), (3, a), (3, b), (3, c), (4, a), (4, b), (4, c), (4, d), (5, a), (5, b), (5, c), (5, d), (5, e)\}\]

- Explanation: Each pair \( (a, b) \) is included if the numeric value of \( a \) (from set \( A \)) matches or exceeds the alphabetic position of \( b \) (from set \( B \)).

 

Question 3: Cartesian Products in Function Graphing

Given the function \( f(x) = x^2 \) from \( \mathbb{R} \) to \( \mathbb{R} \), represent its graph as a subset of \( \mathbb{R} \times \mathbb{R} \) and describe its features.

 

Solution:

- Function Definition: \( f(x) = x^2 \)

- Graph Representation:

  \[\{(x, x^2) | x \in \mathbb{R}\}\]

- Graph Features: The graph is a parabola opening upwards, intersecting the origin. It is symmetric about the y-axis, illustrating that for every positive \( x \), there is a corresponding negative \( x \) that results in the same \( y \)-value.

 

Question 4: Probability with Dice Using Cartesian Products

In a game involving two dice, represent the sample space using a Cartesian product. Calculate the probability of rolling a sum of 7.

 

Solution:

- Sample Space: \( S = \{1, 2, 3, 4, 5, 6\} \times \{1, 2, 3, 4, 5, 6\} \)

- Event for Sum of 7: \( E = \{(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)\} \)

- Probability Calculation:

  \[P(\text{Sum of 7}) = \frac{|E|}{|S|} = \frac{6}{36} =\frac{1}{6}\]

- Explanation: The probability of rolling a sum of 7 is determined by the number of favorable outcomes divided by the total outcomes in the sample space.

 

Question 5: Using Cartesian Products to Define Matrix Relations

Given a matrix \( A \) representing a binary relation on set \( \{1, 2, 3\} \), express this relation as a subset of \( A \times A \) assuming the matrix is:

 

\[A = \begin{bmatrix}0 & 1 & 0 \\1 & 0 & 1 \\0 & 1 & 0 \\\end{bmatrix}\]

 

Solution:

- Relation \( R \) based on matrix \( A \):

  \[R = \{(1, 2), (2, 1), (2, 3), (3, 2)\}\]

- Explanation: The matrix elements \( a_{ij} = 1 \) indicate a relation from \( i \) to \( j \). This relation is non-symmetric except where explicitly shown by the matrix (no diagonal elements are 1).

Domain and Range

Graph of Functions

Graphs of Function

Introduction

The graphs of a function provide much information about the function. All the information we need to draw the graph of a function accurately can be summarized as DR. T. SAM:

• D: Domain
• R: Range
• T: Transformation
• S: Symmetry
• A: Asymptotes
• M: Maximum and Minimum

Now we shall discuss some important graphs\

 

(a) \(f(x)=\frac{2 x}{1+x^{2}}\).\
(i) \(f(x)\) is a odd function,\
(ii) graph is symmetrical about origin,\
(iii) \(f(x)\) attains minimum value\
(iv) Maximum value of the function \(f(x)\) equals 1.\
(v) \(f(0)=0\) and \(|f(x)| \leq 1\) indeed \((1-|x|)^{2} \geq 0\) or \(1+x^{2} \geq|x|\),\
or \(1 \geq \frac{2|\mathrm{x}|}{1+\mathrm{x}^{2}}=|\mathrm{f}(\mathrm{x})|\)\
Since \(f(x) \geq 0\) at \(x \geq 0\) and \(f(1)=1\) in the inter val \([0, \infty)\) the maximum value of the function \(f(x)\) equals 1 , the minimum value being zero.\
Domain : \(-\infty<x<\infty\).\
Range: \(-1 \leq \mathrm{f}(\mathrm{x}) \leq+1\)\

 

(b) \(f(x)=\sin ^{2} x-2 \sin x=(\sin x-1)^{2}-1\)

When \(f(x)\) increases then \(\sin x\) decreases. Similarly, \(f(x)\) decreases when \(\sin x\) increases.\\
When \(\sin \mathrm{x}\) increases\\
\((-\pi / 2 \leq x \leq \pi / 2) \sin x\) decreases\\

\((\pi / 2 \leq x \leq 3 \pi / 2)\).\\

 

(c) \(f(x)=x^{4}-2 x^{2}+3=\left(x^{2}-1\right)^{2}+2\)\\
(i) \(f(x)\) is even function,\\
(ii) \(f(x)\) is symmertric about \(y\)-axis,\\
(iii) \(f(x)\) is attains min.value at \(x= \pm 1\),\\
(iv) \(f(x)\) decreases \(x=0\) to 1 and similarly \(x=0\) to-1,\\
(v) \(f(x)\) increases \(x=1\) to \(\infty\),\\

(vi)f (0) \(=3, D_{f}:-\infty<x<\infty\) and \(R_{f}\) : \(2 \leq y<\infty\).\\

 

(d) \(y=f(x)=\cos ^{-1}(\cos x)\)\\

\(f(x)\) is periodic with period \(2 \pi\).

 

\[ f(x) = \begin{cases} x, & 0 \leq x \leq \pi \quad \text{[from definition of } \cos^{-1} x \text{]} \\ 2\pi - x, & \pi \leq x \leq 2\pi \end{cases} \]

 Let \(2\pi - x = x^{1}\), or \(2\pi - x^{1} = x\), \[\begin{gathered} \pi \leq 2\pi - x^{1} \leq 2\pi \\ -\pi \geq x^{1} - 2\pi \geq -2\pi \\ \pi \geq x^{1} \geq 0 \\ 0 \leq x^{1} \leq \pi \end{gathered}\], \(\cos^{-1} \cos(2\pi - x^{1}) = \cos^{-1} \cos(x^{1}) = x^{1} = 2\pi - x\), Domain: \(-n\pi \leq x \leq n\pi \, [1, 2, \ldots, n]\), Range: \(0 \leq f(x) \leq \pi\).

 

(e) \(y=f(x)=\sqrt{\sin x}\)

Here \(\sin x \geq 0,0 \leq x \leq \pi\)

\[\begin{array}{r} 2\pi \leq x \leq 3\pi \\ 2n\pi \leq x \leq (2n+1)\pi \\ [n = 0, 1, 2, \ldots] \end{array}\]

 

Domain : \(2 \mathrm{n} \pi \leq \mathrm{x} \leq(2 \mathrm{n}+1) \pi[n=0,1,2 \ldots]\)\\
Range : \(0 \leq f(x) \leq 1\)\\

 

(f) \(y=f(x)=x^{1 / \log x}\)

Domain : \(0<x<1\) and \(1<x<\infty\)\\
\(f(x)=x^{1 / \log x}\)\\
\(\left.=x^{\left(\log _{10} 10 / \log _{10} x\right.}\right)=x^{\log _{x} 10}\).\\
\(\therefore \mathrm{f}(\mathrm{x})=10\).\\

 

(g) \(y=\left\{\begin{array}{c}\sin x \text { at }-\pi \leq x \leq 0 \\ 2 \text { at } 0<x \leq 1 \\ 1 /(x-1) \text { at } 1<x \leq 4\end{array}\right.\)\\

 

(h) \(\mathrm{y}=[\mathrm{x}]^{2}\), we have

 

\[\begin{aligned} y & = 0, \text{ when } 0 \leq x < 1 \\ & = 1, \text{ when } 1 \leq x < 2 \\ & = 4, \text{ when } 2 \leq x < 3 \\ & = 9, \text{ when } 3 \leq x < 4 \end{aligned}\]

and so on.\\
Similarly in the case of negative values.\\

 

(i) \(y=x^{2}+[x]^{2}\), we have

\[ y = \begin{cases} x^{2}, & \text{if } x \in [0, 1[ \\ x^{2} + 1, & \text{if } x \in [1, 2[ \\ x^{2} + 4, & \text{if } x \in [2, 3[ \\ x^{2} + 1, & \text{if } x \in [-1, 0[ \end{cases} \]

 

(j) \(y=\left\{\begin{array}{l}-2 \text { at } x>0 \\ 1 / 2 \text { at } x=0 \\ -x^{3} \text { at } x<0\end{array}\right.\)\\

 

(k) \(\begin{array}{rlrl}y & =\operatorname{Cos} x+|\operatorname{Cos} x| \\ y & =\cos x+\cos x & & \cos x \geq 0 \\ y & =\cos x-\cos x & & \cos x<0 \\ y & =\cos 2 x & & -\pi / 2 \leq x \leq \pi / 2 \\ y & =0 & & \pi / 2 \leq x \leq 3 \pi / 2\end{array}\)

(l)\[
\begin{aligned}
(\mathrm{i}) \; & y = |x+2|x, \; & \mathbf{y} & = (x+2)x, \; & x \geq -2, \; \\
& y = -(x+2)x, \; & x \leq -2, \; \\
& y = x^{2} + 2x, \; & \mathbf{y} & = (x+1)^{2} - 1, \; & x \geq -2, \; \\
& y = 1 - (x+1)^{2}, \; & x \leq -2
\end{aligned}
\]

(m) \(y=2|x-2|-|x+1|+x\) at \(x \geq 2\)

\[\begin{aligned} y & = 2(x-2) - (x+1) + x, \; & = 2x - 5, \; & -1 \leq x \leq 2, \; \\ y & = 2(2-x) - (x+1) + x, \; & = -2x + 3, \; & x \leq -1, \; \\ y & = 2(2-x) + (x+1) + x, \; & = 5 \end{aligned}\]

 

(n) \(\mathrm{y}=2^{\mathrm{x}}-2^{-\mathrm{x}}\)

Let us define domain as \((-1,1)\)

\[ y = 2(x-2) - (x+1) + x, \; y = 2x - 5, \; -1 \leq x \leq 2, \; y = 2(2-x) - (x+1) + x, \; y = -2x + 3, \; x \leq -1, \; y = 2(2-x) + (x+1) + x, \; y = 5 \]

Set Theory Basics

Types of Relations (Reflexive, Symmetric, Transitive, Equivalence)

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