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.)