Functions Limits and Continuity

Functions can be added, subtracted and multiplied. They can also be divided where the divisor function does not take the value zero. These operations create new functions.Functions can be added, subtracted and multiplied. They can also be divided where the divisor function does not take the value zero. These operations create new functions.

If f(x) and g(x) are two real valued functions, then for every value of x that belongs to both the domains of f and g, we can define the following functions:

Sum Function: (f + g) (x) = f (x) + g (x)

Difference Function: (f - g) (x) = f(x) - g(x)

Product Function: fg(x) = f(x) g(x)

Domain of these functions are {Domain f} {Domain g}

Quotient Function

Scalar Multiplication Function

(c f) (x) = c.f (x) for all x Domain (f)

Example:

Let f(x) =

g(x) =

{Domain (f) = (0,)

f + g (x) = f(x) + g (x) Domain

[0,1]

[0,1]

[0,1]

Composite Functions

If range(f) C Dom(g), we define the composite function of g and f (gof) by

gof(x) = g [f(x)] for all xA

If range (g) C dom f, we define the composite function (fog) of f and g by

fog (x) = f [g(x)] for all xX

Example:

Let f(x) = x²-1, g(x) = 3x-1

Domain f = R

Domain g = R

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

(range of g C domain of f )

= f(3x-1)

= (3x-1)² - 1

= 9x² + 1 - 6x - 1

= 9x² - 6x

Domain fog = R

Inverse Functions

Let f : AB be a real valued one-one and onto function.

Therefore, we can define a function, (denoted by f^-1) called 'the inverse of f ' as follows:

f^-1 : BA such that

Example:

If f: R R is defined by

f(x) = 5x - 7, find f^-1(x) and f^-1(8).

Suggested answer:

Let f (x) = y, then y = 5x - 7

f is a one-one and onto function.

This is inverse function of f.