# CLASS-11RELATION & FUNCTION - UNDEFINED FUNCTION

Undefined Function

0       ∞

If f(a) is any of the forms -----, -----, (0 X ∞), (∞ X - ∞). 0⁰,

0       ∞

1∞, and ∞⁰, then we say that f(x) is

Not defined at x = a. These forms are called indeterminate forms. Thus,

x² - 4                     0

If, f(x) = ----------, then f(2) = -----, and, therefore, f(x) is not

x – 2                      0

defined at x = 2, similarly, f(x) = x˟ is not defined at x = 0, since f(0) = 0⁰, which is indeterminate

log (cosx)                             π               π         ∞

f(x) = ---------- is not defined at x = ------, since f(----) = ------

sec x                                2               2         ∞

which is indeterminate.

a˟ + 1

Example.1) Examine whether x (--------) is an odd or even function.

a˟ - 1

a˟ + 1                         a⁻˟ + 1         1/a˟ + 1

Ans.) Let, f(x) = x (-------). Then f(-x) = (-x) ------- = (-x) -------

a˟ - 1                         a⁻˟ - 1         1/a˟ - 1

1 + a˟           a˟ + 1

=  (- x) -------- = x . -------- = f(x).

1 - a˟           a˟ - 1

Since, f(x) = f(- x), so it is an even function.         (Ans.)

Example.2) Examine whether the function f(x) = sin [log (x + √(x² + 1)] is an odd or even function.

Ans.) f(- x) = sin [log (- x + √(1 + x²)]

√(1 + x²) + x

= sin [log (√(1 + x²) – x --------------]

√(1 + x²) + x

1

=  sin log (--------------)

x + √(1 + x²)

=  sin log [(x + √1 + x²)⁻¹]

=  sin [- log (x + √1 + x²)]

=  - sin [log (x + √1 + x²)]

=  - f(x)

Since f(-x) = - f(x), the given function is an odd function.  (Ans.)