IIT JEE Main Maths -Unit 1- Functions: one-one, onto, and composition of functions- Study Notes-New Syllabus
IIT JEE Main Maths -Unit 1- Functions: one-one, onto, and composition of functions – Study Notes – New syllabus
IIT JEE Main Maths -Unit 1- Functions: one-one, onto, and composition of functions – Study Notes -IIT JEE Main Maths – per latest Syllabus.
Key Concepts:
Functions
A function is a special type of relation in which every element of the domain is associated with exactly one element in the co-domain.
If \( f : A \rightarrow B \), then for each \( x \in A \), there exists a unique \( y \in B \) such that \( y = f(x) \).
Domain: The set of all possible input values of \( x \).
Co-domain: The set of all possible output values as defined by the function.
Range: The set of actual outputs obtained from the function.
Example: If \( f(x) = x^2 \), \( A = \mathbb{R} \), then
Domain = \( \mathbb{R} \), Co-domain = \( \mathbb{R} \), Range = \( [0, \infty) \).
Real-Valued Functions of a Real Variable
A function \( f: A \rightarrow \mathbb{R} \) is called a real-valued function of a real variable if both the domain \( A \) and the range are subsets of real numbers.
Examples:
- \( f(x) = x + 2 \)
- \( f(x) = \sin x \)
- \( f(x) = \log x \)
Types of Functions
(a) Into Function: If the range of \( f \) is a proper subset of the co-domain, i.e. not every element of co-domain is mapped, the function is into.
Example: \( f: \{1,2,3\} \rightarrow \{a,b,c,d\} \) defined by \( f(1)=a, f(2)=b, f(3)=c \) is into.
(b) Onto Function (Surjective): If every element of the co-domain is mapped by at least one element of the domain, it is onto.
Example: \( f: \mathbb{R} \rightarrow \mathbb{R} \), \( f(x) = x^3 \) is onto, since every real number has a real cube root.
(c) One-to-One Function (Injective): If distinct elements of domain map to distinct elements of co-domain.
Example: \( f(x) = 2x + 3 \) is one-to-one because \( f(x_1) = f(x_2) \Rightarrow x_1 = x_2 \).
(d) Bijective Function: A function which is both one-to-one and onto. Such functions have inverses.
Example (One-to-One):
Show that \( f(x) = 3x – 5 \) is one-to-one.
▶️ Answer / Explanation
If \( f(x_1) = f(x_2) \), then \( 3x_1 – 5 = 3x_2 – 5 \Rightarrow x_1 = x_2 \).
Hence, \( f \) is one-to-one.
Example (Onto):
Show that \( f(x) = x^3 \) is onto from \( \mathbb{R} \) to \( \mathbb{R} \).
▶️ Answer / Explanation
Let \( y \in \mathbb{R} \). For \( f(x) = x^3 \), we can find \( x = \sqrt[3]{y} \).
Since for every \( y \) there exists a real \( x \), \( f \) is onto.
Example (Into Function):
\( f(x) = e^x \), \( f: \mathbb{R} \rightarrow \mathbb{R} \)
▶️ Answer / Explanation
Range of \( e^x \) = \( (0, \infty) \) while co-domain = \( \mathbb{R} \).
Since \( (0, \infty) \) is a proper subset of \( \mathbb{R} \), \( f \) is into.
Algebra of Functions
If \( f \) and \( g \) are two real functions, then:
- Sum: \( (f + g)(x) = f(x) + g(x) \)
- Difference: \( (f – g)(x) = f(x) – g(x) \)
- Product: \( (f \cdot g)(x) = f(x) \cdot g(x) \)
- Quotient: \( \left(\dfrac{f}{g}\right)(x) = \dfrac{f(x)}{g(x)} \), provided \( g(x) \neq 0 \)
Example (Algebra of Functions):
Let \( f(x) = 2x + 3 \) and \( g(x) = x^2 \). Find \( (f + g)(x), (f – g)(x), (f \cdot g)(x), \) and \( \left(\dfrac{f}{g}\right)(x) \).
▶️ Answer / Explanation
\( (f + g)(x) = x^2 + 2x + 3 \)
\( (f – g)(x) = -x^2 + 2x + 3 \)
\( (f \cdot g)(x) = 2x^3 + 3x^2 \)
\( \left(\dfrac{f}{g}\right)(x) = \dfrac{2x + 3}{x^2}, \, x \neq 0 \)
Composite Function
If \( f : A \rightarrow B \) and \( g : B \rightarrow C \), then the composite function \( g \circ f : A \rightarrow C \) is defined as
\( (g \circ f)(x) = g(f(x)) \)
Example (Composite Function):
Let \( f(x) = 2x + 3 \), \( g(x) = x^2 \). Find \( (g \circ f)(x) \) and \( (f \circ g)(x) \).
▶️ Answer / Explanation
\( (g \circ f)(x) = g(f(x)) = (2x + 3)^2 = 4x^2 + 12x + 9 \)
\( (f \circ g)(x) = f(g(x)) = 2x^2 + 3 \)
Important Real Functions
| Type of Function | Definition / Example | Graph |
|---|---|---|
| Absolute Value Function | \( f(x) = |x| \) Range = \( [0, \infty) \) | Graph of V-shape (symmetric about y-axis) |
| Polynomial Function | \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_0 \) Example: \( f(x) = 2x^3 – 3x^2 + 5 \) | Smooth continuous curve depending on degree |
| Rational Function | \( f(x) = \dfrac{p(x)}{q(x)} \), \( q(x) \neq 0 \) Example: \( f(x) = \dfrac{x^2 – 1}{x + 1} \) | Hyperbolic curve with asymptotes |
| Trigonometric Function | \( f(x) = \sin x, \cos x, \tan x, \ldots \) | Wave-like periodic curves |
| Exponential Function | \( f(x) = a^x \), \( a > 0, a \neq 1 \) Range = \( (0, \infty) \) | Rapidly increasing or decreasing curve |
| Logarithmic Function | \( f(x) = \log_a x \), \( a > 0, a \neq 1, x > 0 \) Range = \( \mathbb{R} \) | Slowly increasing curve for \( x > 0 \) |
Example (Absolute Value):
Find range of \( f(x) = |x – 3| + 2 \).
▶️ Answer / Explanation
\( |x – 3| \ge 0 \Rightarrow f(x) \ge 2 \).
Hence, Range = \( [2, \infty) \).
Example (Exponential and Logarithmic):
Find the domain of \( f(x) = \log(x^2 – 9) \).
▶️ Answer / Explanation
Argument of log must be positive: \( x^2 – 9 > 0 \Rightarrow x > 3 \text{ or } x < -3 \).
Domain = \( (-\infty, -3) \cup (3, \infty) \).
Example (Rational Function):
Find domain of \( f(x) = \dfrac{x^2 + 1}{x – 2} \).
▶️ Answer / Explanation
Denominator ≠ 0 ⇒ \( x ≠ 2 \).
Hence, Domain = \( \mathbb{R} – \{2\} \).
Special Real-Valued Functions and Their Properties
1. Signum Function
Defined as \( f(x) = \text{sgn}(x) = \dfrac{x}{|x|} \) for \( x \neq 0 \), and \( f(0) = 0 \).
\[ \text{sgn}(x) = \begin{cases} 1, & \text{if } x > 0 \\ 0, & \text{if } x = 0 \\ -1, & \text{if } x < 0 \end{cases} \]
Domain: \( \mathbb{R} \), Range: \(\{-1, 0, 1\}\).
Example:
Find \( f(-3), f(0), f(5) \) for \( f(x) = \text{sgn}(x) \).
▶️ Answer / Explanation
\( f(-3) = -1, \, f(0) = 0, \, f(5) = 1 \).
2. Greatest Integer Function (GIF)
Denoted by \( f(x) = \lfloor x \rfloor \). It gives the greatest integer less than or equal to \( x \).
Domain: \( \mathbb{R} \), Range: Set of all integers \( \mathbb{Z} \).
Examples: \( \lfloor 4.7 \rfloor = 4 \), \( \lfloor -1.3 \rfloor = -2 \).
The graph is a step-wise constant function with jumps at integer points.
Example:
Evaluate \( \lfloor x \rfloor + \lfloor -x \rfloor \) for \( x = 2.4 \).
▶️ Answer / Explanation
\( \lfloor 2.4 \rfloor = 2 \), \( \lfloor -2.4 \rfloor = -3 \)
\( \therefore \lfloor x \rfloor + \lfloor -x \rfloor = 2 – 3 = -1 \)
3. Fractional Part Function
Defined as \( f(x) = \{x\} = x – \lfloor x \rfloor \).
Domain: \( \mathbb{R} \), Range: \( [0, 1) \).
It represents the fractional portion of a real number.
Example:
Find \( \{ -2.6 \} \).
▶️ Answer / Explanation
\( \lfloor -2.6 \rfloor = -3 \)
\( \{ -2.6 \} = -2.6 – (-3) = 0.4 \)
4. Even and Odd Functions
Even Function: \( f(x) \) is even if \( f(-x) = f(x) \).
Odd Function: \( f(x) \) is odd if \( f(-x) = -f(x) \).
Examples:
- \( f(x) = x^2 \) → even
- \( f(x) = x^3 \) → odd
- \( f(x) = \sin x \) → odd, \( f(x) = \cos x \) → even
5. Increasing and Decreasing Functions
A function \( f(x) \) is:
- Increasing if \( x_1 < x_2 \Rightarrow f(x_1) < f(x_2) \)
- Decreasing if \( x_1 < x_2 \Rightarrow f(x_1) > f(x_2) \)
Example: \( f(x) = x^2 \) is decreasing on \( (-\infty, 0] \) and increasing on \( [0, \infty) \).
6. Inverse of a Function
If \( f: A \rightarrow B \) is a one-to-one and onto (bijective) function, then its inverse \( f^{-1}: B \rightarrow A \) is defined by
\( f^{-1}(y) = x \) if \( f(x) = y \).
Condition: Only bijective functions have inverses.
Example:
Find \( f^{-1}(x) \) if \( f(x) = 3x + 2 \).
▶️ Answer / Explanation
Let \( y = 3x + 2 \). Swap \( x \) and \( y \): \( x = 3y + 2 \).
Solving for \( y \): \( y = \dfrac{x – 2}{3} \).
Thus, \( f^{-1}(x) = \dfrac{x – 2}{3} \).
Notes and Study Materials
- Concepts of Functions
- Concepts of Real Functions
- Concepts of Inverse Trigonometric Function
- Functions Master File
- Functions Revision Notes
- Functions Formulae
- Functions Reference Book
- Functions Past Many Years Questions and Answer
Examples and Exercise
IIT JEE (Main) Mathematics ,”Functions” Notes ,Test Papers, Sample Papers, Past Years Papers , NCERT , S. L. Loney and Hall & Knight Solutions and Help from Ex- IITian
About this unit
Functions; one-one, into and onto functions, the composition of functions.
IITian Academy Notes for IIT JEE (Main) Mathematics – Functions
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