Functions & Graphs — Concept, Formulas & Examples

Sets

INTRODUCTION

sets are a collection of well-defined objects or elements. A set is represented by a capital letter symbol and the number of elements in the finite set is represented as the cardinal number of a set in a curly bracket {…}. For example, set A is a collection of all the natural numbers, such as A = {1,2,3,4,5,6,7,8,…..∞}.

Sets can be represented in 2 forms:

1. Roster Form: All the elements of the set are listed. Example, the set of all odd natural numbers less than 11 is represented as {1, 3, 5, 7, 9}
2. Set Builder Form: Set is described by a characteristising property. Example the set of all odd natural numbers less than 11 is represented as {x ∈ N | x < 12 and x is even}. | means 'such that'

TYPES OF SETS

Empty or Null Set: It has no element present in it. Example: A = {} is a null set.

Finite Set: It has a zero or limited number of elements. Example: A = {1,2,3,4}

Infinite Set: It has an infinite number of elements.Example: A = {x: x is the set of all whole numbers}

Equal Set: Two sets which have the same members.Example: A = {1,2,5} and B={2,5,1}: Set A = Set B

Subset Set: A set ‘A’ is said to be a subset of B if each element of A is also an element of B. Example: A = {1,2}, B = {1,2,3,4}, then A ⊆ B

• Proper subset: B. If A is a subset of B and there is at least one element in B that is not there in A, A is said to be a proper subset of B. This is written as A ⊂ B.
• A ⊄ B means A is not a subset of B.
• A ⊄ B means A is a not a proper subset of B.
• If A is a subset of B we say that B contains A or B is a superset of A. This is written as B ⊇ A

Note: Note: A ⊂ B ⇒ A ⊆ B. But the converse is not true.

Power Set: : If A is any set, then the set of all subsets of A is called the power set of A and is denoted by P(A), i.e., P(A) = {S│S ⊆ A}

Example: If A = {1, 2, 3}, then P(A) = {φ, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}

Universal Set: The set that contains all the elements in a given context is called the Universal Set. It is denoted by µ or U.

BASIC SETS OPERATIONS

UNION OF SETS

: If A and B are two sets, the union of A and B is the set of all those elements which belong to either A or to B or to both A and B. This is denoted by A ∪ B.
A ∪ B = {x │x ∈ A or x ∈ B}

INTERSECTION OF SETS

Let A and B be two sets. The intersection of A and B is the set of all those elements that belong to both A and B. It is denoted by A ∩ B.
A ∩ B = {x | x ∈ A and x ∈ B}

Two sets A and B are said to be disjoint if they have no element in common.
∴ If A and B are disjoint, then A ∩ B = φ.

Example: A = {1, 3, 5} and B = {2, 4, 6}
Here, set A and B are disjoint sets since they have no common element.

COMPLEMENT OF A SET

The complement of set A is the set of all those elements that do not belong to set A, i.e. the complement of a set A is the difference of the universal set and set A and is denoted by A' or A^c.

Example: If µ is the set of natural numbers, A is the set of even natural numbers, then A' is the set of odd natural numbers.

DIFFERNCE OF SETS

The difference of two sets is the set of all elements which are there in one set but not in the other. Let A and B be two sets. A − B is the set of all those elements of A which do not belong to B.

A − B = {x | x ∈ A and x ∉ B}
Similarly, B − A = { x | x ∈ B and x ∉ A}.

Example: A = {1, 2, 3, 4}, B = {3, 4, 8, 10}
A − B = {1, 2}
B − A = {8, 10}

Function & Graphs

Functions and Graphs is a very important chapter from CAT and XAT perspective. You can expect 2 to 3 quesetions from this chapter in CAT. SNAP and NMAT do not ask questions from this chapter.

FUNCTIONS

A function y = f(x) is where x is the input and f(x) is the output.

x is the independent variable here and y is the dependent variable.

Example: f(x) = 2x + 3 is a function whose input is x and (2x + 3) is the output.
If we have to calculate value of f(2), it means we need to calculate the output when input i.e., x = 2. ∴ f(2) = 2 × 2 + 3 = 7.

Similarly, if y = f(x) = x² - 2x + 3, then
f(-1) = (-1)² - 2 × (-1) + 3 = 1 + 2 + 3 = 6.

A function is defined only if for every input (x) there is only one output (f(x) or y).

y = 3x³ + 3x + 5 is a function, but

y² = 2x + 3 is not a function.
because at x = 3, y can be +3 or -3 i.e., there are 2 outputs possible for the same input.

Composite Function

The composition of a function is an operation where two functions say f and g generate a new function say h in such a way that h(x) = g(f(x)). It means here function g is applied to the function of x. So, basically, a function is applied to the result of another function.

Note: f(g(x)) can also be written as fog(x). This means f or g(x).

Piecewise Function

A function which has different definitions depending upon the value of the independent variable is called a Piecewise Function. Thus, the definition of the function changes according to the input.

Example: $\mathrm{f(x)}=\{\begin{array}{ll}\mathrm{x}& \mathrm{x}\ge 0\\ -\mathrm{x}& \mathrm{x}<0\end{array}$

Here, if x is ≥ 0 f(x) = x and
if x < 0 , f(x) = -x

∴ f(2) = 2 but f(-2) = -(-2) = 2

Peiodic Function

A periodic function is a function that repeats itself after regular intervals. Thus, if f(x) = f(x + c), where c is some constant, for all values of x, then f(x) is a periodic function. c is called the period/periodicity of the function.

The most common periodic functions are the trigonometric functions like sin x, cos x etc.

Even Odd Function

A function f(x) is an even if f(-x) = f(x)

Exmaple: f(x) = 2²
Here, f(-x) = (-x)² = x² = f(x)

A function f(x) is a odd if f(-x) = -f(x)

Exmaple: f(x) = 2³
Here, f(-x) = (-x)³ = -x³ = -f(x)

Note: Not all functions are even or odd. Some function may neither be even nor odd.

Inverse Function

If f(x) = y, then inverse function is defined as f^-1y = x.

In a function y = f(x), x is the input and y is the output,
In an inverse function x = f^-1y, y is the input and x is the output.

Example: If f(x) = y = 2x + 3, then
⇒ x = $\frac{\mathrm{(y\; -\; 3)}}{2}$
⇒ f^-1y = g(y) = $\frac{\mathrm{(y\; -\; 3)}}{2}$

We can replace y with x, and hence the inverse function g(x) = $\frac{\mathrm{(y\; -\; 3)}}{2}$

Not all functions have an inverse. Whether or not an inverse exists for a function, can be found using graphs. If the reflection of the graph of a function about the line y = x satisfies the vertical line test, then the inverse exists.