explain functions class 11

1. What is a Function?

A function is a rule that takes every element from one set (called domain) and gives back exactly one element in another set (called codomain).

2. How to Write a Function

We often write a function as
$f(x)$
where f is the name of the function and x is the input.

Example:
$f(x) = 2x + 3$

3. Important Terms

1. Domain – all possible inputs you allow.
2. Range – all actual outputs you really get.
3. Codomain – the set you claim outputs will lie in (range is a subset of codomain).

4. Testing If a Relation Is a Function

A relation is a bunch of ordered pairs $(x, y)$. It is a function only if no $x$ repeats with different $y$.

Graph trick: Vertical Line Test – if any vertical line cuts the graph at more than one point, it is not a function.

5. Types of Elementary Functions (Class 11 syllabus)

1. Identity: $f(x) = x$
2. Constant: $f(x) = k$
3. Polynomial: $f(x) = ax^n + bx^{n-1} + \dots$
4. Rational: $f(x) = \frac{p(x)}{q(x)}$
5. Modulus (Absolute value): $f(x) = |x|$
6. Greatest Integer (Floor): $f(x) = \lfloor x \rfloor$
7. Exponential: $f(x) = a^x$, $a>0$
8. Logarithmic: $f(x) = \log_a x$, $a>0,a\neq1$

6. Composition and Inverse

Composition:
$(f \circ g)(x) = f\big(g(x)\big)$
Inverse: If $f$ is one-one and onto, there exists $f^{-1}$ such that
$f\big(f^{-1}(x)\big)=x$ and $f^{-1}\big(f(x)\big)=x$.

Think of a function like a magic machine.
You drop a number into the machine, it does its secret work, and only one number pops out.

Example:
If the machine rule is “multiply by 2 and add 3”, then putting 4 inside gives:
4 × 2 + 3 = 11.
Every time you feed 4, you always get 11—never anything else.

Key points for a 10-year-old:

1. One input → Only one output.
2. Rule must be clear so you always know what comes out.
3. You can draw it with arrows: 4 → 11, 7 → 17, etc.

Below is a step-by-step checklist to analyse or build any class-11 level function problem.

1. Identify the Formula
   Write it clearly, e.g. $f(x)=2x+5$

2. Find Domain
   • Denominators ≠ 0
   • Even roots need non-negative radicand
   • Log need positive argument.

3. Compute Range (if asked)
   • If simple linear, range is all real numbers
   • Otherwise use algebra or graphical methods.

4. Check Type (one-one, onto)
   • For one-one: assume $f(x_1)=f(x_2)$, show $x_1=x_2$
   • For onto: For any $y$ in codomain, find $x$ in domain.

5. Inverse (if exists)
   Swap $x,y$ and solve.

6. Composition
   Substitute one function into another carefully.

7. Draw Graph
   Helps in visualising domain, range, and nature of function.