   Chapter 10.3, Problem 84E Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203

Solutions

Chapter
Section Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203
Textbook Problem

In Exercises 75-88, identify all singular points of discontinuity of the given function, [HINT: See Example 3.] f ( x ) = { 1 − x      if  x ≤ 1 x + 2     if 1< x < 3 x 2 − 4      if  x ≥ 3

To determine

To calculate: The singular points and points of discontinuity of the function f(x)={1x     if  x1x+2    if  1<x<3x24  if  x3.

Explanation

Given Information:

The function is f(x)={1x     if  x1x+2    if  1<x<3x24  if  x3.

Formula used:

Continuity of closed form function theorem:

Every function that is closed is continuous on its domain.

If limxaf(x) exists and f(a) is defined, then

limxaf(x)=f(a)

The function is discontinuous at point x=a, if either,

a) The limit limxaf(x) does not exist or

b) The limit limxaf(x) does exist but it’s not equal to f(a).

Where, ‘a’ is any real number.

Calculation:

Consider the function f(x)={1x     if  x1x+2    if  1<x<3x24  if  x3.

At x=1 and x=3, the function changes from one closed-form to another. So, the points x=1 and x=3 are the only possible points for discontinuity or singular points.

Check at x=1 if the function is defined or not.

At x=1 the function is 1x. So,

f(1)=1(1)=11=0

At x=1 the function is equal to 0, which is defined.

Hence, it cannot be singular point.

Check at x=3 if the function is defined or not.

At x=3 the function is x24. So,

f(3)=(3)24=94=5

At x=3 the function is equal to 5, which is defined.

Hence, it cannot be singular point.

Thus, there are no singular points.

Now, check for discontinuity at x=1

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