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 xaxis.

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