Functions

Applied Mathematics Class 11 - Chapter 6

1. Introduction to Functions

Function as a Special Relation

A relation $f$ from a set $A$ to a set $B$ is called a function if every element of $A$ has one and only one image in $B$ .

•Every element in set $A$ (Domain) must have an image in set $B$ .
•Unique Image: No element in $A$ can be associated with more than one element in $B$ .
•Many-to-One is allowed: Different $x$ values can have the same $y$ value (e.g., $f(x) = x^2$ ).

"Every function is a relation, but not every relation is a function."

Mapping (Arrow) Diagrams

Mapping diagrams use arrows to show how elements of the Domain (Set A) are paired with elements of the Co-domain (Set B).

Set A

Set B

✅ Is a Function

One-to-One mapping: Every input has a unique output. Perfect!

Vertical Line Test (Graphical Method)

If every vertical line intersects a curve at most once, then the curve represents a function.

Imagine a scanner moving across the graph. If it hits the curve at two points simultaneously, it's not a function!

Vertical Line Test

Scanning x = -4.0

Passed!

For every x, there is exactly one y. This is a function.

"One input must have exactly one output."

2. Domain, Co-domain, and Range

Domain (X)

Range (Y)

Visual Study

Think of Domain as the "footprint" of the curve on the horizontal floor (x-axis), and Range as its "shadow" on the vertical wall (y-axis).

Equation

f(x) = x²/2 - 2

Domain(-∞, ∞)

Range[-2, ∞)

Basic Definitions

The set of all first elements of ordered pairs in

f

is the Domain. The set of all second elements is the Range.

ℹ️

Polynomials

Domain is always

\mathbb{R}

The Zero-Denominator Rule (Domain)

The function $\frac{P(x)}{Q(x)}$ is undefined when $Q(x) = 0$ .

Example: $f(x) = \dfrac{1}{x-1}$

1.Set denominator to zero: $x-1 = 0 \Rightarrow x=1$ .
2.This value makes the fraction 'divide by zero', which is undefined.
3.Domain = $\mathbb{R} \setminus \{1\}$ .

The Horizontal Asymptote (Range)

For $y = \frac{a}{x-b}$ , $y$ can take any value except the one that makes $x$ undefined.

Example: $f(x) = \dfrac{1}{x-1}$

1.Let $y = \dfrac{1}{x-1}$ . Solve for $x$ in terms of $y$ .
2. $x-1 = \dfrac{1}{y} \Rightarrow x = 1 + \dfrac{1}{y}$ .
3.For $x$ to be a real number, the denominator $y$ cannot be $0$ .
4.Range = $\mathbb{R} \setminus \{0\}$ .

The Non-Negative Rule (Square Roots)

The value inside $\sqrt{\dots}$ must be $\geq 0$ .

Example: $f(x) = \sqrt{x-5}$

1.Set expression $\geq 0$ : $x-5 \geq 0 \Rightarrow x \geq 5$ .
2.The function only exists for $x$ values 5 and above.
3.Domain = $[5, \infty)$ .

Quick Reference: Rational

For

f(x) = \dfrac{P(x)}{Q(x)}

, the domain is

\mathbb{R} \setminus \{x : Q(x) = 0\}

Quick Reference: Square Root

For

f(x) = \sqrt{g(x)}

, the domain is

\{x : g(x) \geq 0\}

3. Real-Valued Functions

Algebra of Functions

Addition

(f + g)(x) = f(x) + g(x)

Subtraction

(f - g)(x) = f(x) - g(x)

Multiplication

(fg)(x) = f(x) \cdot g(x)

Division

(f/g)(x) = \dfrac{f(x)}{g(x)}, g(x) \neq 0

4. Standard Functions & Their Graphs

f(x) = x

f(x) = c

f(x) = x² + 2x + 1

f(x) = x³

Check the "Step" and "Signum" functions - notice their distinctive jumps!

5. Specialized Applied Math Functions

📊 Exponential Functions: $2^x, 3^x, (0.5)^x$

Exponential functions grow or decay at constant rates. As the exponent increases, the value grows exponentially.

📌 Key Observations:

✓ All curves pass through the point (0, 1) because $a^0 = 1$
✓ When base > 1: exponential growth (curves rise)
✓ When 0 < base < 1: exponential decay (curves fall)
✓ Larger bases grow faster (compare $2^x$ and $3^x$ )

📈 Logarithmic Functions: $log_2(x), log_3(x), log_{10}(x)$

Logarithmic functions are the inverse of exponential functions. They grow slowly and are useful for measuring scales that span many orders of magnitude.

📌 Key Observations:

✓ All curves pass through the point (1, 0) because $log_a(1) = 0$
✓ Logarithmic functions grow very slowly compared to exponentials
✓ Smaller base = steeper curve (compare $log_2$ and $log_{10}$ )
✓ Undefined for $x \le 0$ (logarithms only exist for positive values)
✓ These are inverse functions of exponentials

6. Practical Applications

Cost, Revenue & Profit

Cost Function

C(x) = F + v(x)

(Fixed + Variable Cost)

Revenue Function

R(x) = p \cdot x

(Price

\times

Quantity)

Profit Function

P(x) = R(x) - C(x)

Demand & Supply

Demand: $p = f(q)$ (Generally decreasing); Supply: $p = g(q)$ (Generally increasing).

Market Equilibrium

Occurs where Demand = Supply (Point of Intersection).

Loading Knowledge...

Preparing your next chapter

1. Introduction to Functions

Function as a Special Relation

Mapping (Arrow) Diagrams

✅ Is a Function

Vertical Line Test (Graphical Method)

Vertical Line Test

Passed!

2. Domain, Co-domain, and Range

Visual Study

Basic Definitions

Polynomials

The Zero-Denominator Rule (Domain)

The Horizontal Asymptote (Range)

The Non-Negative Rule (Square Roots)

Quick Reference: Rational

Quick Reference: Square Root

3. Real-Valued Functions

Algebra of Functions

4. Standard Functions & Their Graphs

f(x) = x

f(x) = c

f(x) = x² + 2x + 1

f(x) = x³

5. Specialized Applied Math Functions

📊 Exponential Functions: 2x,3x,(0.5)x2^x, 3^x, (0.5)^x2x,3x,(0.5)x

📌 Key Observations:

📈 Logarithmic Functions: log2(x),log3(x),log10(x)log_2(x), log_3(x), log_{10}(x)log2​(x),log3​(x),log10​(x)

📌 Key Observations:

6. Practical Applications

Cost, Revenue & Profit

Cost Function

Revenue Function

Profit Function

Demand & Supply

Market Equilibrium

📝 Practice

Concept Checks

📂 Real-World Functions

Domain Tip

Graph Tip

Practice Scenarios

🔥 CBSE Exam Priority Questions

Important Questions (Applied Maths)

Loading Knowledge...

1. Introduction to Functions

Function as a Special Relation

Mapping (Arrow) Diagrams

✅ Is a Function

Vertical Line Test (Graphical Method)

Vertical Line Test

Passed!

2. Domain, Co-domain, and Range

Visual Study

Basic Definitions

Polynomials

The Zero-Denominator Rule (Domain)

The Horizontal Asymptote (Range)

The Non-Negative Rule (Square Roots)

Quick Reference: Rational

Quick Reference: Square Root

3. Real-Valued Functions

Algebra of Functions

4. Standard Functions & Their Graphs

f(x) = x

f(x) = c

f(x) = x² + 2x + 1

f(x) = x³

5. Specialized Applied Math Functions

📊 Exponential Functions: 2x,3x,(0.5)x2^x, 3^x, (0.5)^x2x,3x,(0.5)x

📌 Key Observations:

📈 Logarithmic Functions: log2(x),log3(x),log10(x)log_2(x), log_3(x), log_{10}(x)log2​(x),log3​(x),log10​(x)

📌 Key Observations:

6. Practical Applications

Cost, Revenue & Profit

Cost Function

Revenue Function

Profit Function

Demand & Supply

Market Equilibrium

📝 Practice

📊 Exponential Functions: $2^x, 3^x, (0.5)^x$

📈 Logarithmic Functions: $log_2(x), log_3(x), log_{10}(x)$

📊 Exponential Functions: $2^x, 3^x, (0.5)^x$

📈 Logarithmic Functions: $log_2(x), log_3(x), log_{10}(x)$