Sets: A set is a group of objects wherein each object is called a member of the group. A set can be represented using curly brackets so a set containing all even numbers till 10 is the set {2, 4, 6, 8, and 10}.

Relations: A relation is just a relationship between sets of information. A well-defined relation is called a function.

Function: A function is a rule that assigns to each element in a set exactly one element called .

Function can be easily defined with the help of the concept of mapping. Let X and Y be any two non-empty sets. "A function from X to Y is a rule or correspondence that assigns to each element of set X, one and only one element of set Y". Let the correspondence be 'f' then mathematically we write f:X Y where y = f(x), x X and y Y. We say that 'y' is the image of 'x' under 'f' (or x is the pre image of y). Two things should always be kept in mind: (i) A mapping f: X Y is said to be a function if each element in the set X has its image in set Y. It is possible that a few elements in the set Y are present which are not the images of any element in set X. (ii) Every element in set X should have one and only one image. That means it is impossible to have more than one image for a specific element in set X. Functions can't be multi-valued (A mapping that is multi-valued is called a relation from X to Y)

Range & Domain of a function :

set is called the domain of the function. The number is the value of at and is read .The range of is the set of all possible values of as varies throughout the domain.

The Vertical Line Test: : A curve in the plane is the graph of a function if and only if no vertical line intersects the curve more than once The reason for the truth of the Vertical Line Test can be seen in Figure . If each vertical line intersects a curve only once, at , then exactly one functional value is defined by . But if a line intersects the curve twice, at and , then the curve cant represent a function because a function cant assign two different values to .

Continuity A function f(x) is said to be continuous at x= a if limxa- f(x)= limxa+ f(x)=f(a) i.e. L.H.L. = R.H.L. = f(a)= value of the function at a i.e. limxa f(x) = f(a) If f(x) is not continuous at x= a, we say that f(x) is discontinuous at x=a. For the function to be continuous at any point x=a, the function must be defined at that point and limiting values of f(x) when x approaches a, is equal to f(a). Continuity means the function should not have any break or sudden jump at any point in the given domain.

(i) Continuous function in [a, b]

(ii) Discontinuous function at x=c

(iii) Discontinuous at x=c So the condition for continuity if function at x=a can be defined as L.H.L.=f(a)= R.H.L. i.e. limxa- f(x) = limxa+ f(x) = f(a) i.e. left hand limit is equal to the value of the function at that point and is equal to the right hand limit of the function at that point.

f(x) will be discontinuous at x=a in any of the following cases: limxa- f(x) and limxa+ f(x) exist but are not equal. limxa- f(x) and limxa+ f(x) exist and are equal but not equal to f(a). f(a) is not defined At least one of the limits does not exist.

Absolute Value Function The function defined as:

is called an absolute value function. Note : x2 = |x| x R

The graph of an absolute value function is shown in the figure given above. Its properties are: (i) An absolute value function is an even function (ii) It is strictly increasing in [0, ) and strictly decreasing in (-, 0]. Illustration: Draw the graph of the following functions. (a) y = |x - 1| + |x - 4| (b) y = |sin x| (c) y = sin |x| (a) Note: x - 1 = 0 => x = 1 and x - 4 = 0 => x = 4 i.e. y changes its definition at x =1 and x = 4. y = |x - 1| + |x - 4| let - < x < 1 y = -(x - 1) - (x - 4) = -2x + 5 Now, let 1 < x < 4 y = (x - 1) - (x - 4) = 3 Again, Let 4 < x y = (x - 1) + (x - 4) = 2x - 5

(b) y = |sin x| y > 0 x R

(c) y = sin |x| x > 0, y = sin x x < 0, y = sin (-x) = -sinx

One-to-One Function: A function f from A to B is called one-to-one if whenever f (a) = f (b) then a = b. No element of B is the image of more than one element in A. These functions are also termed as injective. Given any y there is only one x that can be paired with the given y.

"One-to-One"

Invertible Function: A function that is one-to-one as well as onto is called invertible. Let f be a function whose domain is the set X, and whose range is the set

Y. Then f is invertible if there exists a function g with domain Y and range X, with the property: f(x) = y iff g(y) = x.

If the function f is invertible, the function g is unique. This g is called the inverse of f and written as g = f -1

Method to Find Inverse of a Function Note : A function and its inverse are always symmetric with respect to the line y = x. Illustration: Let f : R R defined by f(x) = (ex-e-x)/2 . Find f-1 (x). Solution: We have f(f-1(x)) = x => (ef-1(x) - e-f-1(x))/2 = x => e2f-1(x) - 2xef-1(x) -1 = 0 => ef-1(x) = x + (x2 +1). But negative sign is not possible because L.H.S. is always positive. Thus ef-1(x) = x + (x2 +1) . Hence, f-1(x) = log(x + (x2 +1)) .

(i) Many-one and onto (surjective).

(ii) One-one (injective) and into.

(iii) One-one (injective) and onto (surjective) i.e. Bijective.

(iv) and (v) are not functions.

Even Function: Let f(x) be a real valued function of a real variable. Then f is even if the following equation holds for all x and -x in the domain of f:

f(x) = f(-x)

Geometrically, the graph of an even function is symmetric with respect to the y-axis.

Odd Function: Again, let f(x) be a real valued function of a real variable. Then f is odd if the following equation holds for all x and -x in the domain of f:

f (-x) = - f(x) or f(x) + f(-x) =0.

Increasing & Decreasing function:

Composite Functions

Another useful combination of two functions f and g is the composition of these two functions. Let f : X Y and g : Y Z be two functions.

We define a function h : X Z by setting h(x) = g(f(x). To obtain h(x), we first take the f-image f(x), of an element x in X so that f(x) Y, which is the domain of g(x) and then take the g-image of f(x), that is, g(f(x)), which is an element of Z. The scheme is shown in the figure.

The function h, defined above, is called the composition of f and g and is written gof. Thus (gof)(x) = g(f(x)). Domain of gof = {x : x in domain f, f(x) in domain g}.

e.g. Let f : R R be a function defined by f(x) = x2 + 4 and g[0, ) R be a function defined by g(x) = x. Then gof(x) = g(f(x)) = (x2 + 4). Domain of gof = R. Thus we have gof : R R defined by (gof)(x) = (x2 + 4). Similarly, we shall have fog : [0, ) R defined by (fog)(x) = x + 4. Note that (gof)(x) (fog)(x).

Illustration: Two functions are defined as under:

Find fog and gof.

Solution: (fog)(x) = f(g(x))

Let us consider, g(x) < 1 :

(i) x2 < 1, -1 < x < 2 => -1 < x < 1, -1 < x < 2 => -1 < x < 1

(ii) x2 + 2 < 1, 2 < x < 3 => x < -1, 2 < x < 3 => x =

Let us consider, 1 < g(x) < 2,

(iii) 1 < x2 < 2, -1 < x < 2

=> x [-2, -1) (1,2] , -1 < x < 2 => 1 < x < 2

(iv) 1 < x+2 < 2, 2 < x < 3 => -1 < x < 0, 2 < x < 3, x =

Let us consider -1 < f(x) < 2 :

(i) -1 < x+1 < 2, x < 1 => -2 < x < 1, x < 1 => -2 < x < 1

(ii) -1 < 2x+1 < 2, 1 < x < 2 => -1 , x < , 1 < x < 2 => x=

Let us consider 2 < f(x) < 3:

(iii) 2 < x+1 < 3 , x < 1 => x < 2 , x < 1 => x = 1

(iv) 2 < 2x+1 < 3, 1 < x < 2 => 1 < 2x < 2, 1 < x < 2

=> < x < 1 , 1 < x < 2 => x =

Rules of Inequality :

a < b => either a < b or a = b

a < b and b < c => a < c a < b => a + c < b + c c R a < b => -a > -b i.e. inequality sign reverses if both sides are multiplied by a

negative number a < b and c < d => a + c < b + d and a - d < b - c a < b => ma < mb if m > 0 and ma > mb if m < 0 0 < a < b => ar < br if r > 0 and ar > br if r < 0 (a+(1/2)) > 2 a > 0 and equality holds for a = 1 (a+(1/2)) < -2 a > 0 and equality holds for a = -1

Solutions of linear inequalities in one variable : Rule 1 Equal numbers may be added to (or subtracted from) both sides of an inequality without affecting the sign of inequality. Rule 2 Both sides of an inequality can be multiplied (or divided) by the same positive number. But when both sides are multiplied or divided by a negative number, then the sign of inequality is reversed. Example 1: Solve (5 2x)/3 (x/6) 5 Example 2 : Solve 7x + 3 < 5x + 9. Show the graph of the solutions on number line. Example 3 : Solve (3x 4)/2 (x + 1)/4 1. Show the graph of the solutions on number line. Graphical solution of linear inequalities in two variables : 1. Graph the solution to 2x 3y < 6. 2. Graph the solution to this system of inequalities.

3.Graph the solution to this system of inequalities.

4. Solve this system by graphing.

Absolute Value Inequalities : 1. Solve for x: |3 x 5| < 12. 2. Solve this disjunction for x: |5 x + 3| > 2. 3. Solve for x: |2 x + 11| < 0. 4. Solve for x: 7|3 x + 2| + 5 > 4.