Real Functions and Their Properties
A function f from a non-empty set A to a non-empty set B associates to each element x in A with a unique element in B which is denoted by f (x). The set A is called domain of the function f, the set B is called co-domain of f, f (x) is called value of the function f at x and the set {f (x): x ∈ A} is called range of f (or the image of A). f: A → B denotes the function f from A to B, in fact f (x) is the value of the function f at x. f (x) is also called the image of x under the function f.
Functions whose domain and co-domain both are subsets of R, the set of real numbers, are called real valued functions of a real variable i.e. a real function.
Greatest Integer Function (Step or Floor Function)
Let x ∈ ℝ be any real number. [x] denotes the greatest integer less than or equal to x. e.g. [3.01] = 3, [−3.99] = −4 etc.
Fractional Part
We have seen that x ≥ [x]. The difference between the number ‘x’ and its integral value ‘[x]’ is called the fractional part of x and is symbolically denoted as {x}.
Thus, {x} = x − [x]
e.g. if x = 4.92 then [x] = 4
and {x} = 0.92.
We can check that fractional part of any number is always non-negative and less than one.
If x is an integer ⇒ x = [x] ⇒ {x} = 0
⇒ {[x]} = 0
Exponential Function
If \( a \) is a positive real number, then the function defined by f(x) = ax, x ∈ R is called an exponential function to the base a.
Domain f = R; Range (f) = set of positive numbers.
Observations
- a) The graph is above the x-axis.
- b) for a > 1, as we move from left to right, the graph rises above, and for 0 < a < 1, as we move from left to right the graph falls and it looks to touch the x-axis as we approaches to infinity.
If a = e, where e is an irrational number lying between 2 and 3, then the exponential function takes the form f (x) = ex, x ∈ R.
Domain (f) = R
Range (f) = the set of real positive numbers.
The curve is an increasing curve. It passes through (0, 1).
Note:
- For \( a > 1 \):
- If \( -\infty < x \leq 0 \), then \( 0 < f(x) \leq 1 \);
- If \( 0 < x < \infty \), then \( 1 < f(x) < \infty \)
- For \( 0 < a < 1 \):
- If \( 0 \leq x < \infty \), then \( 0 < f(x) \leq 1 \);
- If \( -\infty < x < 0 \), then \( 1 < f(x) < \infty \).
Logarithmic Function
Observations
Logarithmic Function
If a > 0 and a ≠ 1, then the function f : R+ → R f : R+→ R defined by y = loga xy = loga x iff ay= x is called the logarithmic function.
Domain (f) = R+; Range (f) = R
Observations
| 0 < a < 1 | a > 1 |
|---|---|
| • loga x is a decreasing function | • loga x is an increasing function |
| • limx→∞ loga x = −∞ | • limx→∞ loga x = ∞ |
| • limx→0 loga x = ∞ | • limx→0 loga x = −∞ |
| • sign of loga x is positive for 0 < x < 1 and sign of loga x is negative for x > 1 | • sign of loga x is negative for 0 < x < 1 and sign of loga x is positive for x > 1. |
An Important special case of logarithmic function as f (x) = loge x whose graph is
Domain (f) = R+ ; Range (f) = R.
Observations
Curve is increasing and meets x-axis at (1, 0)
Note:
• logba1 ≥ logba2 ⇒ { a1 ≥ a2 > 0 if b > 1, 0 < a1 ≤ a2 if 0 < b < 1 }
Signum Function
The function f: ℜ → ℜ defined by
is called the signum function. The domain of the signum function is ℜ and the range is the set {−1, 0, 1}.
Note: The points (0, 1) and (0, –1) are not included in the graph where as the point (0, 0) is included.
Alternate ways the express signum function are
Even and Odd Function
If f : X → Y is a real valued function such that for all x ∈ D ⇒ – x ∈ D (where D is the domain of f) and if f (–x) = f (x) for all x ∈ D then f is said to be an even function and if f (–x) = – f (x), then f is said to be odd function. Even functions are symmetric about the y-axis and odd functions are symmetric about the origin.
Note:
- A function can be even or odd or neither.
- Every function defined in symmetric interval D(i.e. x∈D⇒−x∈Dx∈D⇒−x∈D) can be expressed as the sum of an even and an odd function.
Composition of two Functions
Let f and g be any two real functions such that the domain of f includes the range of g. Then it makes sense to talk of f(g(x)) for each x in the domain X of g. Thus f(g(x)) is a unique element associated to x, (x ∈ X) in this way. As such it defines a function with domain X. We denote the function by f ∘ g and call it composite of f and g. If f : Y → Z and g : X → Y, then f ∘ g : X → Z is defined by the rule
(f ∘ g)(x) = f(g(x)), for all x ∈ X.
One-One Function (injective Function)
A function f : A → B is said to be one-one or injective function if different elements of set A have different f images in set B. Thus no two elements of set A can have the same f image. In other words f(x1) = f(x2) Þ x1 = x2.
e.g. f : R → R given by f (x) = 2x + 3 is one-one.
Many-One Function
A function f : A → B is called many one if at least one element in the set B is the f image of more than one elements of A.
e.g. f : R → R given by f (x) = x2 + 2 is many-one.
Methods to Determine One-One and Many-One
(i) Let x1, x2 ∈Domain of f and if x1 ≠ x2 ⇒ f(x1) ≠ f(x2) for every x1, x2 in the domain, then f is one-one else many-one.
(ii) Conversely if f(x1) = f(x2) ⇒ x1 = x2 for every x1, x2 in the domain, then f is one-one else many-one.
(iii) A function is one-one if and only if no line parallel to the x-axis meets the graph of the function at more than one point.
Onto or (surjective function)
The function f : A → B is said to be an onto function if every element of B is image of atleast one-element of A. i.e., for each y ∈ B, there exists atleast one x ∈ A such that f(x) = y, then f is an onto function. Range of f = co-domain of f(B)
Into function
If the function f : A → B is such that there is atleast one element of B which is not the image of any element of A, then f is called an into function.
Onto function is also called Surjective function and a function which is both one-one and onto is called Bijective function.
Range of f ⊂ co-domain of f.
e.g. f : R→R where f(x) = sinx is into.
f : R→R where f(x) = ax³ + b is onto where a ≠ 0, b ∈ R.
Note that a function will either be onto or into.
Methods to Determine Function Types
Methods to Determine whether a Function is Onto or Into
(i) If Range = codomain, then f is onto. If range is a proper subset of codomain, then f is into.
(ii) Solve f(x) = y for x, say x = g(y).
Now if g(y) is defined for each y ∈ codomain and g(y) ∈ domain of f for all y ∈ codomain, then f(x) is onto. If this requirement is not met by at least one value of y in codomain, then f(x) is into.
Remark:
An into function can be made onto by redefining the co-domain as the range of the original function.
Inverse Function
Let us consider a one-one onto function with domain A and range B. Let y ∈ B.
This member y ∈ B arises from one and only one member x ∈ A such that f(x) = y, as the function is one-one.
Thus we can define a new function say 'g' such that
g(y) = x ⟺ f(x) = y
we can also denote g by f-1.
In this case domain of f-1 = range of 'f'.
and range of f-1 = domain of 'f'.
Only one-one onto functions are invertible. All one-one onto functions are strictly monotonic in nature, hence sufficient condition for the existence of invertibility of an onto function y = f(x) is that it must be strictly monotonic. If a function increases or decreases then its inverse also increases or decreases accordingly.
Methods of Finding Inverse of a Function
(i) If you are asked to check whether the given function y = f(x) is invertible, you need to check that y = f(x) is one-one and onto.
(ii) If you are asked to find the inverse of a bijective function f(x), you do the following:
If f-1 be the inverse of f, then
f-1of(x) = fof-1(x) = x (always)
Apply the formula of f on f-1(x) and use the above identity to solve for f-1(x).
Formula and Concepts
Mathematical Function Definitions
1. A mapping f: X → Y is said to be a function if each element in the set X has its image in set Y. It is possible that a few elements in the set Y are present, which are not the images of any element in set X.
2. Every element in set X should have one and only one image. That means it is impossible to have more than one image for a specific element in set X. Functions can't be multi-valued (A mapping that is multi-valued is called a relation from X to Y).
3. Set 'X' is called domain of the function 'f'. Set 'Y' is called the co-domain of the function 'f'.
4. Set of images of different elements of set X is called the range of the function 'f'.
5. The function 'h' defined above is called the composition of f and g and is denoted by gof. Thus (gof)x = g(f(x)). Clearly Domain (gof) = {x : x ∈ Domain (f), f(x) ∈Domain(g)}
7. f(x) - f(-x) = 0 even function
8. f(x) + f(-x) = 0 odd function
Frequently Asked Questions
A function in mathematics is a special relationship between two sets of numbers where each input value (called the domain) corresponds to exactly one output value (called the range). Think of a function as a mathematical machine that takes an input, processes it according to a specific rule, and produces a unique output.
Essential Components of a Function:
| Component | Definition | Example |
|---|---|---|
| Input (x) | The independent variable or domain value | x = 3 |
| Rule | The mathematical operation or formula | f(x) = 2x + 1 |
| Output (y) | The dependent variable or range value | y = 7 |
Real-world Example: Consider a vending machine function. You input money (domain), and the machine gives you exactly one specific snack (range) based on your selection. You can't get two different snacks for the same input combination.
Function Notation:
f(x) = y
where f represents the function name, x is the input, and y is the output
Understanding functions is crucial for CBSE students as they form the foundation for advanced mathematical concepts including calculus, algebra, and mathematical modeling. Functions help us describe relationships between quantities in physics, economics, and engineering applications.
Explaining functions to beginners requires using simple analogies and visual representations that connect to everyday experiences. The most effective approach is to start with familiar concepts and gradually introduce mathematical terminology.
Step-by-Step Explanation Method:
- Start with Analogies: Use familiar machines like coffee makers, calculators, or smartphone apps
- Introduce the Input-Output Concept: Show how each input produces exactly one output
- Use Visual Representations: Draw simple diagrams with arrows showing relationships
- Provide Concrete Examples: Use simple mathematical examples before abstract concepts
- Address Common Misconceptions: Clarify why some relationships are not functions
Teaching Example - The Phone Book Function:
Input: Person's name → Output: Their phone number
This works as a function because each person has exactly one phone number listed. If someone had multiple numbers listed, it wouldn't be a function.
Common Beginner-Friendly Analogies:
| Analogy | Input | Process | Output |
|---|---|---|---|
| Vending Machine | Money + Selection | Internal mechanism | Specific product |
| Recipe | Ingredients | Cooking instructions | Finished dish |
| Calculator | Numbers + Operation | Mathematical processing | Result |
Teaching Tip: Always emphasize the "one input, one output" rule. This is the fundamental concept that distinguishes functions from other mathematical relationships.
For CBSE students, it's important to connect these basic concepts to the curriculum progression from Class 11 to Class 12, where functions become increasingly important for understanding limits, derivatives, and integrals. Starting with solid conceptual understanding prevents confusion in advanced topics.
Here are several clear examples of functions that demonstrate the core concept of unique input-output relationships, ranging from simple mathematical expressions to real-world applications relevant to CBSE students.
Mathematical Function Examples:
Example 1: Linear Function
f(x) = 2x + 3
For any input x, multiply by 2 and add 3:
- f(1) = 2(1) + 3 = 5
- f(2) = 2(2) + 3 = 7
- f(3) = 2(3) + 3 = 9
Example 2: Quadratic Function
g(x) = x²
Square the input value:
- g(2) = 2² = 4
- g(-3) = (-3)² = 9
- g(0) = 0² = 0
Real-World Function Examples:
| Situation | Input | Function Rule | Output | Example |
|---|---|---|---|---|
| Temperature Conversion | Celsius (C) | F = (9/5)C + 32 | Fahrenheit (F) | 25°C → 77°F |
| Area of Circle | Radius (r) | A = πr² | Area (A) | r = 5 → A = 25π |
| Simple Interest | Time (t) | I = Prt | Interest (I) | t = 2 years → I = 2Pr |
CBSE Curriculum Examples by Grade:
Class 11 Functions:
- f(x) = |x| (Absolute value function)
- f(x) = [x] (Greatest integer function)
- f(x) = √x (Square root function)
Class 12 Functions:
- f(x) = eˣ (Exponential function)
- f(x) = ln(x) (Natural logarithm function)
- f(x) = sin(x), cos(x) (Trigonometric functions)
These examples show how functions progress from simple arithmetic operations to complex mathematical relationships. Understanding these foundational examples helps students recognize function patterns in physics formulas, chemistry equations, and real-world problem-solving scenarios throughout their CBSE education.
The seven basic functions form the foundation of mathematical analysis and are essential building blocks for understanding more complex mathematical concepts. These functions appear throughout the CBSE curriculum from Class 11 onwards and are crucial for calculus, algebra, and mathematical modeling.
The 7 Basic Functions:
| Function Type | General Form | Key Characteristics | Graph Shape | CBSE Grade |
|---|---|---|---|---|
| Constant Function | f(x) = c | Output never changes | Horizontal line | Class 11 |
| Linear Function | f(x) = mx + b | Constant rate of change | Straight line | Class 11 |
| Quadratic Function | f(x) = ax² + bx + c | U-shaped or inverted U | Parabola | Class 11 |
| Cubic Function | f(x) = ax³ + bx² + cx + d | S-shaped curve | Cubic curve | Class 12 |
| Square Root Function | f(x) = √x | Only positive outputs | Half parabola | Class 11 |
| Absolute Value Function | f(x) = |x| | Always non-negative | V-shaped | Class 11 |
| Reciprocal Function | f(x) = 1/x | Hyperbolic shape | Two branches | Class 12 |
Detailed Characteristics:
Linear Function Properties:
- Domain: All real numbers (-∞, ∞)
- Range: All real numbers (-∞, ∞)
- Slope: Constant value 'm'
- Applications: Distance-time relationships, temperature conversion
Quadratic Function Properties:
- Domain: All real numbers (-∞, ∞)
- Range: Depends on 'a' value and vertex
- Vertex: Point of maximum or minimum
- Applications: Projectile motion, profit maximization
CBSE Examination Focus: These seven functions are frequently tested in various forms including graphing, domain-range identification, function composition, and real-world application problems. Students should master both algebraic manipulation and graphical interpretation of each type.
Understanding these basic functions enables students to tackle complex mathematical problems, recognize patterns in scientific data, and prepare for advanced topics like limits and derivatives in Class 12. Each function type has unique properties that make it suitable for modeling different types of real-world relationships.
What are the different types of functions in mathematics?
Functions can be classified in multiple ways based on different mathematical properties. Understanding these classifications helps students identify function characteristics quickly and choose appropriate solution methods for various mathematical problems in the CBSE curriculum.
Classification by Mathematical Form:
| Category | Types | Examples | Key Features |
|---|---|---|---|
| Algebraic Functions | Polynomial, Rational, Radical | f(x) = x², f(x) = 1/x, f(x) = √x | Involve basic operations |
| Transcendental Functions | Exponential, Logarithmic, Trigonometric | f(x) = eˣ, f(x) = ln(x), f(x) = sin(x) | Cannot be expressed algebraically |
| Special Functions | Step, Absolute, Greatest Integer | f(x) = |x|, f(x) = [x] | Have unique properties |
Classification by Mapping Properties:
One-to-One (Injective) Functions:
- Each output corresponds to exactly one input
- Passes horizontal line test
- Example: f(x) = 2x + 1
- Has an inverse function
Onto (Surjective) Functions:
- Every element in codomain is mapped
- Range equals codomain
- Example: f(x) = x³ (for real numbers)
- Covers entire target set
Many-to-One Functions:
- Multiple inputs map to same output
- Example: f(x) = x² (both 2 and -2 map to 4)
- Most common type in basic mathematics
- Does not have inverse function
Classification by Behavior:
| Property | Definition | Test Method | Example |
|---|---|---|---|
| Even Function | f(-x) = f(x) | Symmetric about y-axis | f(x) = x², f(x) = cos(x) |
| Odd Function | f(-x) = -f(x) | Symmetric about origin | f(x) = x³, f(x) = sin(x) |
| Periodic Function | f(x + T) = f(x) | Repeats after period T | f(x) = sin(x), f(x) = cos(x) |
| Monotonic Function | Always increasing or decreasing | Check derivative sign | f(x) = eˣ, f(x) = ln(x) |
CBSE Application Strategy: Understanding function classifications helps in solving inverse function problems, analyzing symmetry properties, and determining domain-range relationships. These concepts are particularly important for Class 12 calculus applications.
Mastering function classification enables students to quickly identify appropriate solution strategies, predict function behavior, and understand the mathematical relationships that appear throughout physics, chemistry, and engineering applications in the CBSE curriculum.
A linear function is a mathematical relationship where the rate of change between variables remains constant. It creates a straight line when graphed and represents one of the most fundamental function types in mathematics, extensively used throughout the CBSE curriculum from Class 11 onwards.
Standard Form and Components:
f(x) = mx + b
where m = slope and b = y-intercept
| Component | Symbol | Meaning | Effect on Graph |
|---|---|---|---|
| Slope | m | Rate of change | Steepness and direction |
| Y-intercept | b | Starting value when x = 0 | Vertical position |
| Independent Variable | x | Input value | Horizontal axis |
| Dependent Variable | f(x) or y | Output value | Vertical axis |
Key Properties:
Mathematical Properties:
- Domain: All real numbers (-∞, ∞)
- Range: All real numbers (-∞, ∞)
- Continuity: Continuous everywhere
- Derivative: f'(x) = m (constant)
Real-World Applications in CBSE Context:
| Subject | Application | Linear Function | Interpretation |
|---|---|---|---|
| Physics | Uniform Motion | s = ut + s₀ | Distance vs. time |
| Chemistry | Temperature Conversion | F = (9/5)C + 32 | Celsius to Fahrenheit |
| Economics | Cost Function | C = mx + b | Total cost vs. quantity |
| Mathematics | Simple Interest | A = P + Prt | Amount vs. time |
Graphical Characteristics:
- Always forms a straight line
- Slope determines steepness: positive (rising), negative (falling), zero (horizontal)
- Y-intercept shows where line crosses vertical axis
- X-intercept found by setting f(x) = 0
CBSE Examination Strategies:
Common Question Types:
- Finding slope and intercepts from graphs or equations
- Writing equations given two points or slope-intercept
- Solving systems of linear equations
- Word problems involving linear relationships
- Graphing linear inequalities
Linear functions serve as building blocks for understanding more complex mathematical concepts in Class 12, including differential calculus where the derivative of any function represents instantaneous rate of change, generalizing the constant rate concept from linear functions.
What is a quadratic function in mathematics?
A quadratic function is a polynomial function of degree 2 that creates a distinctive U-shaped curve called a parabola when graphed. These functions are fundamental in CBSE mathematics, appearing prominently in Class 11 algebra and extending into Class 12 calculus applications.
Standard Forms:
| Form | Equation | Best Used For | Key Information |
|---|---|---|---|
| Standard Form | f(x) = ax² + bx + c | General analysis | Coefficients a, b, c |
| Vertex Form | f(x) = a(x - h)² + k | Finding vertex | Vertex at (h, k) |
| Factored Form | f(x) = a(x - r₁)(x - r₂) | Finding roots | Roots at r₁ and r₂ |
Key Components and Properties:
Coefficient 'a' determines:
- Direction: a > 0 (opens upward), a < 0 (opens downward)
- Width: |a| > 1 (narrow), |a| < 1 (wide)
- Vertex type: a > 0 (minimum), a < 0 (maximum)
Finding the Vertex:
Vertex x-coordinate: x = -b/(2a)
Vertex y-coordinate: y = f(-b/(2a))
Domain and Range Analysis:
| Property | Value | Explanation |
|---|---|---|
| Domain | (-∞, ∞) | All real numbers can be input |
| Range (a > 0) | [k, ∞) | Minimum value is k (vertex y-coordinate) |
| Range (a < 0) | (-∞, k] | Maximum value is k (vertex y-coordinate) |
Solving Quadratic Equations:
Solution Methods:
- Factoring: When expression factors easily
- Quadratic Formula: x = [-b ± √(b² - 4ac)] / 2a
- Completing the Square: Converting to vertex form
- Graphical Method: Finding x-intercepts
Discriminant Analysis:
Discriminant: Δ = b² - 4ac
| Discriminant Value | Number of Real Roots | Graph Characteristics |
|---|---|---|
| Δ > 0 | Two distinct real roots | Crosses x-axis at two points |
| Δ = 0 | One repeated real root | Touches x-axis at vertex |
| Δ < 0 | No real roots | Does not touch x-axis |
Real-World Applications:
CBSE Physics Applications:
- Projectile Motion: h(t) = -½gt² + v₀t + h₀
- Kinetic Energy: KE = ½mv²
- Area Relationships: A = πr² (circle area)
Business Mathematics:
- Profit Optimization: P(x) = -ax² + bx - c
- Cost Functions: Including fixed and variable costs
- Revenue Models: Price-demand relationships
Understanding quadratic functions is essential for CBSE students as they bridge algebraic thinking with calculus concepts. The parabolic relationships modeled by quadratics appear frequently in physics problems, optimization scenarios, and form the foundation for understanding more complex polynomial behaviors in advanced mathematics.
Understanding the distinction between relations and functions is fundamental for CBSE students, as it establishes the foundation for all advanced mathematical concepts. While all functions are relations, not all relations qualify as functions due to specific mapping requirements.
Definitions and Core Differences:
| Aspect | Relation | Function |
|---|---|---|
| Definition | Any set of ordered pairs | Special relation with unique mapping |
| Mapping Rule | One input can have multiple outputs | Each input has exactly one output |
| Mathematical Notation | R = {(x,y) | condition} | f: A → B where f(x) = y |
| Domain Requirement | Can have unused domain elements | Every domain element must be mapped |
| Vertical Line Test | May fail test | Must pass test |
Visual Recognition Methods:
Vertical Line Test:
- Draw vertical lines through the graph
- If any vertical line intersects the graph more than once, it's not a function
- If every vertical line intersects at most once, it is a function
Examples with Analysis:
| Example | Type | Reason | Mathematical Expression |
|---|---|---|---|
| Student ID → Student Name | Function | Each ID maps to one name | f(ID) = Name |
| Person → Phone Number | Relation (not function) | One person may have multiple numbers | R = {(person, phone)} |
| x² + y² = 25 | Relation (not function) | One x-value gives two y-values | Circle equation |
| y = x² + 3 | Function | Each x gives exactly one y | f(x) = x² + 3 |
Set Theory Representation:
Relation Example:
A = {1, 2, 3}, B = {4, 5, 6}
R = {(1,4), (1,5), (2,6), (3,4)}
This is a relation but not a function because 1 maps to both 4 and 5.
Function Example:
A = {1, 2, 3}, B = {4, 5, 6}
f = {(1,4), (2,5), (3,6)}
This is a function because each element in A maps to exactly one element in B.
Common CBSE Question Patterns:
Identification Questions:
- Given a set of ordered pairs, determine if it's a function
- Analyze graphs using the vertical line test
- Find domain and range of relations vs. functions
- Real-world scenario classification
Practical Applications in CBSE Context:
| Subject | Relation Example | Function Example |
|---|---|---|
| Physics | Position vs. time (oscillation) | Distance vs. time (uniform motion) |
| Chemistry | Atomic number ↔ isotopes | Atomic number → element name |
| Geography | Country ↔ languages spoken | Capital city → country |
Mastering the relation-function distinction enables students to understand when mathematical operations are well-defined, predict solvability of equations, and recognize when inverse relationships exist. This foundational knowledge becomes crucial in Class 12 for understanding limits, continuity, and differentiability concepts.