A function is a rule which indicates an operation to perform. A function is a relation between input and output. For example, there is a system, which finds the square of the given input. That means the output is a square of the given input.

This can be represented by output = (input)2

f(x) = x2

where x is input and f(x) is output. Here f is called the function of x, which is defined as f(x) = x2.

Example 1: If f(x) = 2x2 – 2x + 1, find f(–1)

Substitute –1 in place of x.

f(–1) = 2(–1)2 – 2(–1) + 1

### Odd and Even Functions

Odd function: A function f is said to be odd if it changes sign when the sign of the variable is changed.

If f(–x) = – f(x)

For example: f(x) = sin x ; 0 ≤ x ≤ 2π is a odd function.

Even function: A function f is said to be an even function if it doesn’t change sign when the sign of the variable is changed.

if f(–x) = f(x)

For example: f(x) = x4 + x2 and g(x) = cos x are even functions.

There are many functions which are neither odd nor even i.e it is not necessary for a function to be either even of to be odd.

### Composite Functions

A composite function is the function of another function. If f is a function from A in to B and g is a function from B in to C, then their composite function denoted by (g o f) is a function from A in to C defined by

(g o f)(x) = g[f(x)]

For example, if f(x) = 2x, and g(x) = x + 2, Then

(gof)(x) = g[f(x)] = g(2x) = 2x + 2

(fog)(x) = f[g(x)] = f(x + 2) = 2(x + 2) = 2x + 4

This shows that it is not necessary that (fog)(x) = (gof)(x).

Example 2: Let a function fn+1(x) = fn(x) + 3. If f2(2) = 4. Find the value of f6(2).

fn+1(x) = fn(x) + 3

f3(2) = f2(2) + 3 = 7

f4(2) = f3(2) + 3 = 10

f5(2) = f4(2) + 3 = 13

f6(2) = f5(2) + 3 = 16

Alternate Method:

Since the function is increasing with constant value.

So, f6(2) = f2(2) + 3( 6 – 2) = 4 + 12 = 16

f(x) = k

Domain: R

Range: k

### Modulus Function

f(x) = |x|

f(x) = –x when x < 0
= x when x ≥ 0

Domain: R

Range: R+

f(x) = x

Domain: R

Range: R

f(x) = [x]

Domain: R

Range: Integer

f(x) = 1/x

Domain: R – {0}

Range: R – {0}

### Graph Transformations

1. y = f(x) + a is the same as the graph y = f(x), shifted upwards by a units.

2. y = f(x - a) shifts the graph a units to the right.

3. y = f(ax) is a stretch with scale factor 1/a parallel to the x-axis.

4. y = a.f(x) is a stretch with scale factor a parallel to the y-axis.