A function f : A → B is one-one if:

Distinct elements of A have distinct images. One-one means no two elements share an image.

Every element of B has a pre-image

The function f : R → R given by f(x) = x² is:

Neither one-one nor onto. f(2) = f(−2), and negative numbers have no pre-image.

The function f : R → R, f(x) = 3x − 1 is:

One-one and onto. A linear function with non-zero slope is a bijection on R.

If f(x) = x + 1 and g(x) = 2x, then (gof)(x) is:

2x + 2. (gof)(x) = g(x + 1) = 2(x + 1) = 2x + 2.

Maharashtra State Board · HSC Commerce · Class 11 (Std XI)

Class 11 Maths and Statistics Part 1 Chapter 2: Functions

Short notes, key concepts, quick revision and practice MCQs with answers

A function is a special relation that gives every element of the domain exactly one image. The chapter covers domain, co-domain and range, one-one and onto functions, evaluating functions, composite and inverse functions, and standard types such as polynomial, rational, modulus, signum, greatest integer, exponential and logarithmic functions.

Key concepts: Functions

Function: A relation f from A to B that associates every element x of A with a unique element y of B, written y = f(x). y is the image of x and x is a pre-image of y.
Domain, co-domain and range: A is the domain, B is the co-domain and the set of all images f(x) is the range. The range is always a subset of the co-domain.
One-one and onto: One-one (injective): distinct elements have distinct images. Onto (surjective): every element of B has a pre-image, i.e. range = co-domain. Into: at least one element of B has no pre-image.
Composite function: If f : A → B and g : B → C, then (gof)(x) = g(f(x)). In general gof ≠ fog.
Inverse function: If f is one-one and onto, its inverse f⁻¹ exists and f(x) = y ⇔ f⁻¹(y) = x. Domain of f = range of f⁻¹ and range of f = domain of f⁻¹.
Fundamental functions: Constant, identity f(x) = x, polynomial, rational p(x)/q(x) with q(x) ≠ 0, modulus |x|, signum (−1, 0, 1) and greatest integer [x], the greatest integer not exceeding x.
Exponential and logarithmic functions: f(x) = aˣ (a > 0, a ≠ 1) and its inverse f(x) = logₐ x (x > 0). logₐ(mn) = logₐ m + logₐ n, logₐ(m/n) = logₐ m − logₐ n, logₐ mⁿ = n logₐ m, log_b a = log a / log b.

Quick revision: important points

Every function is a relation, but every relation is not a function.
f(x) = x² from R to R is neither one-one nor onto; f(x) = ax + b (a ≠ 0) from R to R is one-one and onto.
Domain of 1/(x − a) is R − {a}; domain of √(x − a) is [a, ∞); domain of log x is (0, ∞).
[2.7] = 2 and [−2.3] = −3.
logₐ 1 = 0 and logₐ a = 1.
To find f⁻¹: write y = f(x), solve for x in terms of y, then replace y by x.

Practice MCQs with answers

If f(x) = 2x + 3, then f(2) is:
1. 5
2. 7
3. 9
4. 10
Show answer
Answer: B. 7
f(2) = 2(2) + 3 = 7.
A function f : A → B is one-one if:
1. Every element of B has a pre-image
2. Distinct elements of A have distinct images
3. Range equals domain
4. A = B
Show answer
Answer: B. Distinct elements of A have distinct images
One-one means no two elements share an image.
The function f : R → R given by f(x) = x² is:
1. One-one and onto
2. One-one but not onto
3. Onto but not one-one
4. Neither one-one nor onto
Show answer
Answer: D. Neither one-one nor onto
f(2) = f(−2), and negative numbers have no pre-image.
The function f : R → R, f(x) = 3x − 1 is:
1. One-one and onto
2. One-one only
3. Onto only
4. Neither
Show answer
Answer: A. One-one and onto
A linear function with non-zero slope is a bijection on R.
If f(x) = x + 1 and g(x) = 2x, then (gof)(x) is:
1. 2x + 1
2. 2x + 2
3. x + 2
4. 2x² + 2x
Show answer
Answer: B. 2x + 2
(gof)(x) = g(x + 1) = 2(x + 1) = 2x + 2.

All Class 11 Maths and Statistics chapters

Class 11 Maths and Statistics notes: all chapters