# Functions

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) = x^{2}

where x is input and f(x) is output. Here f is called the function of x, which is defined as f(x) = x^{2}.

**Example 1: If f(x) = 2x ^{2} - 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) = x^{4} + x^{2} 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 f _{n+1}(x) = f_{n}(x) + 3. If f_{2}(2) = 4. Find the value of f_{6}(2).**

f_{n+1}(x) = f_{n}(x) + 3

f_{3}(2) = f_{2}(2) + 3 = 7

f_{4}(2) = f_{3}(2) + 3 = 10

f_{5}(2) = f_{4}(2) + 3 = 13

f_{6}(2) = f_{5}(2) + 3 = 16

Alternate Method:

Since the function is increasing with constant value.

So, f_{6}(2) = f_{2}(2) + 3( 6 – 2) = 4 + 12 = 16

### Constant Function

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^{+}

### Identity function

f(x) = x

Domain: R

Range: R

### Greatest Integer Function or Step Function

f(x) = [x]

Domain: R

Range: Integer

### Reciprocal Function

f(x) = 1/x

Domain: R - {0}

Range: R - {0}

### Graph Transformations

- y = f(x) + a is the same as the graph y = f(x), shifted upwards by a units.
- y = f(x - a) shifts the graph a units to the right.
- y = f(ax) is a stretch with scale factor 1/a parallel to the x-axis.
- y = a.f(x) is a stretch with scale factor a parallel to the y-axis.