HOW TO CHECK CONTINUITY OF MODULUS FUNCTION

About "How to Check Continuity of Modulus Function"

How to Check Continuity of Modulus Function :

Here we are going to how to examine the continuity of the modulus function.

To know the points to be remembered in order to decide whether the function is continuous at particular point or not, you may look into the page " How to Check Continuity of a Function If Interval is not Given "

How to Check Continuity of Modulus Function - Examples

Question 1 :

Examine the continuity of the following

|x + 2| + |x - 1|

Solution :

Since we have a absolute value function, we have to split into piece wise function and check its continuity.

 x + 2  =  0x  =  -2 x - 1  =  0x  =  1 f(x)  =  -x - 2 - x + 1  =  -2x - 1     If x < -2

f(x)  =  x + 2 - x + 1  =  3      If -2 ≤ x < 1

f(x)  =  x + 2 + x - 1  =  2x + 1   If x ≥ 1

From the above piece wise function, we have to check if it is continuous at x = -2 and x = 1

 lim x->-2- f(x) =  -2(-2) - 1  =  4 - 1    =  3 ---(1) lim x->-2+ f(x) =  3 ---(2)

Since left hand limit and right hand limit are equal for -2, it is continuous at x = -2.

 lim x->1- f(x) =  3 lim x->1+ f(x) =  2x + 1  =  2(1) + 1  =  3

Since left hand limit and right hand limit are equal for 1, it is continuous at x = 1.

Hence the function is continuous for all x ∈ R.

Question 2 :

Examine the continuity of the following

|x - 2| / |x + 1|

Solution :

x - 2  =  0          x + 1  =  0

x  =  2    x  =  -1 f(x)  =  (x-2)/(x+1)     If x < -1

f(x)  =  -[(x-2)/(x+1)]     If -1 < x < 2

f(x)  =  (x-2)/(x+1)      If x > 2

From the above piece wise function, we have to check if it is continuous at x = -1 and x = 2

lim x->-1- f(x)

=  (-1-2)/(-1+1)

=  -3/0

=  -

The function is not continuous at x = -1.

Now we need to check if it is continuous at x = 2

 lim x->2- f(x)   =    -[(2-2)/(x+1)]  =  0 lim x->2- f(x)   =    [(2-2)/(x+1)]  =  0

It is continuous at x = 2.

Hence the function is continuous for  all x ∈ R - {-1}.

Question 3 :

Examine the continuity of the following

cot x + tan x

Solution :

Let f(x)  =  cot x + tan x

tan x is not defined at x = π/2 and odd multiples of π/2.

cot x is not defined at x = π/2 and even multiples of π/2.

Hence the function is continuous for all ∈ R - nπ/2 and n ∈ z. After having gone through the stuff given above, we hope that the students would have understood, "How to Check Continuity of Modulus Function"

Apart from the stuff given in "How to Check Continuity of Modulus Function" if you need any other stuff in math, please use our google custom search here. WORD PROBLEMS

HCF and LCM  word problems

Word problems on simple equations

Word problems on linear equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6