📐 Mathematics-1A: Functions
Complete Guide for EAPCET & JEE Mains | 15 Years Teaching Experience
📋 Table of Contents (16 Sections)
- Basics of Functions
- Types of Functions
- Polynomial Functions
- Rational Functions
- Exponential Functions
- Logarithmic Functions
- Trigonometric Functions
- Special Functions
- Function Properties
- Operations on Functions
- Transformations of Functions
- Comparison Summary
- Problem-Solving Strategies
- Practice Problems
- Essential Formulas
- Exam Tips
1. BASICS OF FUNCTIONS
Definition of a Function
Function f: A → B such that each element of A is associated with exactly ONE element of B.
f: A → B, where x ∈ A ⟹ f(x) ∈ B
- Domain: Set of all INPUT values (A)
- Codomain: Set of possible outputs (B)
- Range: Set of all ACTUAL outputs
Domain and Range Table
| Type | Form | Domain | Range |
|---|---|---|---|
| Polynomial | aₙxⁿ+…+a₀ | ℝ | ℝ or [m,∞) |
| Rational | p(x)/q(x) | ℝ-{0s of q} | Varies |
| √x | √x | [0,∞) | [0,∞) |
| logₐx | logₐx | (0,∞) | ℝ |
| aˣ | aˣ | ℝ | (0,∞) |
2. TYPES OF FUNCTIONS
2.1 One-One (Injective)
f(x₁)=f(x₂) ⟹ x₁=x₂
Different inputs give different outputs
2.2 Onto (Surjective)
Range = Codomain
Every element in codomain has at least one pre-image
2.3 Bijective
Both One-One AND Onto
⭐ Only bijective functions have inverses!
3. POLYNOMIAL FUNCTIONS
General Form
f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀
3.1 Linear: f(x) = mx + c
- Domain: ℝ | Range: ℝ
- Slope: m | y-intercept: c
- Type: Bijective ✓
- Graph: Straight line
3.2 Quadratic: f(x) = ax² + bx + c
- Domain: ℝ
- Vertex: (-b/2a, f(-b/2a))
- Axis: x = -b/2a
- a > 0: Opens UP, Range = [min, ∞)
- a < 0: Opens DOWN, Range = (-∞, max]
3.3 Cubic: f(x) = ax³ + bx² + cx + d
- Domain: ℝ | Range: ℝ
- Type: Bijective ✓
- Turning points: ≤ 2
4. RATIONAL FUNCTIONS
f(x) = p(x)/q(x), where q(x) ≠ 0
Vertical Asymptote: x = a where q(a) = 0
Horizontal Asymptote (deg p = deg q):
y = (lead coeff p)/(lead coeff q)
Example: f(x) = 1/x
- Domain: ℝ – {0}
- Range: ℝ – {0}
- V. Asymptote: x = 0
- H. Asymptote: y = 0
- Type: Odd, Bijective
5. EXPONENTIAL FUNCTIONS
f(x) = aˣ where a > 0, a ≠ 1
Properties
- Domain: ℝ
- Range: (0, ∞)
- Always positive: aˣ > 0
- a⁰ = 1
- Bijective ✓
Laws of Exponents
- aᵐ·aⁿ = aᵐ⁺ⁿ
- aᵐ/aⁿ = aᵐ⁻ⁿ
- (aᵐ)ⁿ = aᵐⁿ
- (ab)ⁿ = aⁿbⁿ
Case 1: a > 1 (Growth)
- Increasing function
- Passes: (0, 1)
- H.A.: y = 0 as x → -∞
- Examples: 2ˣ, eˣ
Case 2: 0 < a < 1 (Decay)
- Decreasing function
- Passes: (0, 1)
- H.A.: y = 0 as x → +∞
- Examples: (1/2)ˣ, e⁻ˣ
6. LOGARITHMIC FUNCTIONS
f(x) = logₐx where a > 0, a ≠ 1
Definition: y = logₐx ⟺ aʸ = x
Properties
- Domain: (0, ∞)
- Range: ℝ
- logₐ(1) = 0
- logₐ(a) = 1
- Bijective ✓
Laws
- logₐ(mn) = logₐm + logₐn
- logₐ(m/n) = logₐm – logₐn
- logₐ(mⁿ) = n·logₐm
- logₐx = ln x/ln a
Case 1: a > 1
- Increasing
- Passes: (1, 0)
- V.A.: x = 0
Case 2: 0 < a < 1
- Decreasing
- Passes: (1, 0)
- V.A.: x = 0
⭐ INVERSE: aˣ ↔ logₐx
logₐ(aˣ) = x
a^(logₐx) = x
Graphs are reflections about y = x
7. TRIGONOMETRIC FUNCTIONS
7.1 Sine: f(x) = sin(x)
- Domain: ℝ | Range: [-1, 1]
- Period: 2π
- Odd: sin(-x) = -sin(x)
- sin(0)=0, sin(π/2)=1, sin(π)=0
7.2 Cosine: f(x) = cos(x)
- Domain: ℝ | Range: [-1, 1]
- Period: 2π
- Even: cos(-x) = cos(x)
- cos(0)=1, cos(π/2)=0, cos(π)=-1
7.3 Tangent: f(x) = tan(x)
- Domain: ℝ – {(2n+1)π/2}
- Range: ℝ
- Period: π
- Odd: tan(-x) = -tan(x)
- V.A.: x = (2n+1)π/2
8. SPECIAL FUNCTIONS
8.1 Modulus: f(x) = |x|
|x| = { x if x ≥ 0
-x if x < 0 }
- Domain: ℝ | Range: [0, ∞)
- Even: |-x| = |x|
- V-shaped graph
- NOT bijective
8.2 Greatest Integer: f(x) = [x]
[x] = greatest integer ≤ x
- Domain: ℝ | Range: ℤ
- [3.7]=3, [-2.5]=-3
- Step function
8.3 Signum: f(x) = sgn(x)
sgn(x) = { 1 if x > 0
0 if x = 0
-1 if x < 0 }
- Domain: ℝ | Range: {-1, 0, 1}
- Odd function
- Step function
8.4 Fractional Part: f(x) = {x}
{x} = x – [x]
- Domain: ℝ | Range: [0, 1)
- {3.7}=0.7, {-2.5}=0.5
- Period: 1
9. FUNCTION PROPERTIES
9.1 Even & Odd
EVEN
f(-x) = f(x)
Symmetric about Y-axis
Ex: x², cos(x), |x|
ODD
f(-x) = -f(x)
Symmetric about origin
Ex: x³, sin(x), tan(x)
9.2 Monotonic
Increasing
x₁ < x₂ ⟹ f(x₁) ≤ f(x₂)
Decreasing
x₁ < x₂ ⟹ f(x₁) ≥ f(x₂)
Strictly Incr
x₁ < x₂ ⟹ f(x₁) < f(x₂)
9.3 Periodic
f(x + T) = f(x) for all x
Examples: sin(2π), tan(π), {x}(1)
9.4 Bounded
Bounded above: ∃ M, f(x) ≤ M
Example: sin(x) bounded by 1
Bounded below: ∃ m, f(x) ≥ m
Example: x² bounded below by 0
10. OPERATIONS ON FUNCTIONS
10.1 Arithmetic Operations
(f+g)(x) = f(x)+g(x), Domain: D₁∩D₂
(f-g)(x) = f(x)-g(x), Domain: D₁∩D₂
(f·g)(x) = f(x)·g(x), Domain: D₁∩D₂
(f/g)(x) = f(x)/g(x), Domain: D₁∩D₂-{g(x)=0}
10.2 Composition
(f∘g)(x) = f(g(x))
Apply g FIRST, then f
Note: (f∘g) ≠ (g∘f) usually
10.3 Inverse Function
f⁻¹(f(x)) = x
f(f⁻¹(y)) = y
Only bijective functions have inverses!
Finding Inverse
- Replace f(x) with y
- Solve for x in terms of y
- Swap x and y
- Result is f⁻¹(x)
11. TRANSFORMATIONS OF FUNCTIONS
11.1 Vertical Shift
g(x) = f(x) + c
c > 0: UP | c < 0: DOWN
11.2 Horizontal Shift
g(x) = f(x – h)
h > 0: RIGHT | h < 0: LEFT
11.3 Vertical Stretch/Compress
g(x) = a·f(x)
|a| > 1: STRETCH | 0 < |a| < 1: COMPRESS
11.4 Horizontal Stretch/Compress
g(x) = f(bx)
|b| > 1: COMPRESS | 0 < |b| < 1: STRETCH
11.5 Reflections
About x-axis
g(x) = -f(x)
About y-axis
g(x) = f(-x)
About y=x
g(x) = f⁻¹(x)
12. COMPARISON SUMMARY TABLE
| Function | Domain | Range | Type |
|---|---|---|---|
| f(x)=x | ℝ | ℝ | Identity, Odd, Bijective |
| f(x)=x² | ℝ | [0,∞) | Even, Parabola, NOT bijective |
| f(x)=x³ | ℝ | ℝ | Odd, Cubic, Bijective |
| f(x)=√x | [0,∞) | [0,∞) | Increasing, Bijective |
| f(x)=1/x | ℝ-{0} | ℝ-{0} | Odd, Hyperbola, Bijective |
| f(x)=|x| | ℝ | [0,∞) | Even, V-shaped, NOT bijective |
| f(x)=aˣ | ℝ | (0,∞) | Exponential, Bijective, Positive |
| f(x)=logₐx | (0,∞) | ℝ | Logarithmic, Bijective, Inverse aˣ |
| f(x)=sin(x) | ℝ | [-1,1] | Periodic, Odd, Period 2π |
| f(x)=cos(x) | ℝ | [-1,1] | Periodic, Even, Period 2π |
13. PROBLEM-SOLVING STRATEGIES
13.1 Finding Domain – Key Restrictions
- Denominator ≠ 0: 1/(x-2) requires x ≠ 2
- Square root: √x requires x ≥ 0
- Even roots: ⁿ√x (n even) requires x ≥ 0
- Logarithm: ln(x) requires x > 0
- Inverse trig: sin⁻¹(x) requires -1 ≤ x ≤ 1
- Tangent: tan(x) undefined at x = (2n+1)π/2
13.2 Finding Range – Methods
Method 1: Algebraic
- Solve y = f(x) for x in terms of y
- Find restrictions on y from domain of x
Method 2: Calculus
- Find critical points: f'(x) = 0
- Find max/min values
- Check endpoints and limits
13.3 Quick Identification
Example Problems:
- f(x) = ³√(x²): Domain ℝ, Range [0,∞) ✓
- f(x) = x²/(x²+1): Domain ℝ, Range [0,1) ✓
- f(x) = √(x²-4): Domain (-∞,-2]∪[2,∞) ✓
- f(x) = √(|x|-1): Domain (-∞,-1]∪[1,∞) ✓
13.4 Composite Function Domain
Find domain of (f∘g)(x):
- Find domain of g(x)
- Find range of g(x)
- Check which range values are in domain of f(x)
- Combine all conditions
14. PRACTICE PROBLEMS WITH SOLUTIONS
Problem 1: Domain and Range
Q: Find domain and range of f(x) = √(4 – x²)
Solution:
For domain: 4 – x² ≥ 0
x² ≤ 4
-2 ≤ x ≤ 2
Domain = [-2, 2]
When x = 0: f(0) = 2 (maximum)
When x = ±2: f(±2) = 0 (minimum)
Range = [0, 2] ✓
Problem 2: Injective Function
Q: Is f(x) = x³ – 3x one-one on ℝ?
Solution:
f'(x) = 3x² – 3 = 3(x²-1) = 3(x-1)(x+1)
f'(x) = 0 at x = 1 and x = -1
Function increases, then decreases, then increases
NOT monotonic
Answer: NOT one-one ✗
Problem 3: Finding Inverse
Q: Find f⁻¹(x) if f(x) = (2x+1)/(x-3)
Solution:
y = (2x+1)/(x-3)
y(x-3) = 2x+1
xy – 3y = 2x + 1
xy – 2x = 3y + 1
x(y-2) = 3y + 1
x = (3y+1)/(y-2)
f⁻¹(x) = (3x+1)/(x-2) ✓
Problem 4: Composition
Q: If f(x)=x+2 and g(x)=x²-1, find (f∘g)(x) and (g∘f)(x)
Solution:
(f∘g)(x) = f(g(x)) = f(x²-1)
= (x²-1) + 2 = x² + 1 ✓
(g∘f)(x) = g(f(x)) = g(x+2)
= (x+2)² – 1
= x² + 4x + 4 – 1
= x² + 4x + 3 ✓
15. ESSENTIAL FORMULAS REFERENCE
| Topic | Formula | When Used |
|---|---|---|
| Vertex (Quadratic) | (-b/2a, f(-b/2a)) | Parabola turning point |
| Axis of Symmetry | x = -b/2a | Parabola line of symmetry |
| Discriminant | Δ = b² – 4ac | Nature of roots |
| Change of Base | logₐx = log_b(x)/log_b(a) | Convert log bases |
| Exp-Log Inverse | a^(logₐx) = x, logₐ(aˣ) = x | Simplify expressions |
| Pythagorean | sin²(x) + cos²(x) = 1 | Trig simplification |
| Sec-Tan | 1 + tan²(x) = sec²(x) | Trig identities |
| Csc-Cot | 1 + cot²(x) = csc²(x) | Trig identities |
16. IMPORTANT EXAM TIPS & TRICKS
✓ Domain Finding Tips:
- Check ALL denominators – must be ≠ 0
- Check ALL even roots – must be ≥ 0
- Check ALL logarithms – argument must be > 0
- For composite: combine using AND (∩)
- Write answer in standard form using intervals
✓ Range Finding Tips:
- Use y = f(x) and solve for x: check y restrictions
- For bounded functions: always have finite range
- For periodic: range repeats
- For rational: beware of horizontal asymptotes
- Use calculus if algebraic method fails
✓ Function Properties to Remember:
- ⭐ BIJECTIVE = ONE-ONE + ONTO (for inverse)
- EVEN: symmetric about y-axis
- ODD: symmetric about origin
- PERIODIC: pattern repeats with period T
- Check: if f(x)=f(-x) then EVEN
- Check: if f(x)=-f(-x) then ODD
✓ Composition Order (IMPORTANT!):
- (f∘g)(x) means “f of g of x” – apply g FIRST
- Domain of (f∘g) ⊆ domain of g
- Range of (f∘g) ⊆ range of f
- Usually (f∘g) ≠ (g∘f)
✓ Inverse Function Tips:
- Only bijective functions have inverses
- Domain of f⁻¹ = Range of f
- Range of f⁻¹ = Domain of f
- f⁻¹(f(x)) = x and f(f⁻¹(x)) = x
- Graphs of f and f⁻¹ are reflections about y=x
✓ Quick Checks in Exam:
- Verify domain by checking all restrictions
- Verify range by checking min/max values
- Check special points: f(0), f(1), f(-1)
- Check asymptotes for rational/exponential/log
- Always write final answer clearly
