Question

Using the definition of bounded function, show whether the following func-

tions are bounded or not:
(a) f(x) = x+1/x-1 on (-1,1)
(b) g(x) = sin (1/x) on (0,0)

Answer 1

A bounded function is one where the absolute value of the function is less than some real number M at every point in its domain. Function (a) is unbounded due to a discontinuity at x = 1, and function (b) is not given a valid interval but would be unbounded near 0 due to oscillations.

Step-by-step explanation:

The definition of a bounded function is a function f(x) where there exists some real number M such that absolute value of f(x) is less than or equal to M for all x in its domain. To determine if the given functions are bounded, we need to analyze each function within its given domain.

(a) The function f(x) = x + 1/x - 1 on the interval (-1,1) has a discontinuity at x = 1. This causes the function to approach infinity as x approaches 1 from the left, which means f(x) is unbounded on this interval.

(b) The question seems to have a mistake, as g(x) = sin (1/x) on (0,0) is not a valid interval since the start and end points are the same. We need an interval with different start and end points to determine the boundness of the function. Assuming it should've been an interval from 0 to some positive number, sin(1/x) would oscillate infinitely as x approaches 0, making it unbounded on any interval including 0.