Ex 5.1, 34 - Find all points of discontinuity f(x) = |x| - |x+1| - Algebra of continous functions

Slide46.JPG
Slide47.JPG Slide48.JPG Slide49.JPG Slide50.JPG Slide51.JPG

  1. Class 12
  2. Important Question for exams Class 12
Ask Download

Transcript

Ex 5.1, 34 Find all the points of discontinuity of f defined by ๐‘“ ๐‘ฅ๏ทฏ= ๐‘ฅ๏ทฏ โ€“ ๐‘ฅ+1๏ทฏ. When ๐‘ฅโ‰ฅ0 ๐‘“ ๐‘ฅ๏ทฏ = ๐‘ฅโˆ’ ๐‘ฅ+1๏ทฏ โˆ’x โˆ’ x + 1 = x โˆ’ x โˆ’ 1 = ๐‘ฅโˆ’๐‘ฅ+1 = x โˆ’ x โˆ’ 1 = โˆ’1 When ๐‘ฅ<โˆ’1 ๐‘“ ๐‘ฅ๏ทฏ = โˆ’๐‘ฅโˆ’ โˆ’ ๐‘ฅ+1๏ทฏ๏ทฏ = โˆ’๐‘ฅ+๐‘ฅ+1 = 1 When โˆ’1โ‰ค๐‘ฅ<0 ๐‘“ ๐‘ฅ๏ทฏ = โˆ’๐‘ฅโˆ’ ๐‘ฅ+1๏ทฏ = โˆ’๐‘ฅโˆ’๐‘ฅ+1 = โˆ’2๐‘ฅ+1 Now ๐‘“ ๐‘ฅ๏ทฏ= 1 ๐‘–๐‘“ ๐‘ฅโ‰คโˆ’1๏ทฎโˆ’2๐‘ฅโˆ’1 ๐‘–๐‘“ โˆ’1โ‰ค๐‘ฅ<0๏ทฎโˆ’1 ๐‘–๐‘“ ๐‘ฅโ‰ฅ0๏ทฏ๏ทฏ Checking continuity Case 1 At x = 0 A function is continuous at x = 0 if L.H.L = R.H.L = ๐‘“ 0๏ทฏ i.e. lim๏ทฎxโ†’ 0๏ทฎโˆ’๏ทฏ๏ทฏ ๐‘“ ๐‘ฅ๏ทฏ= lim๏ทฎxโ†’ 0๏ทฎ+๏ทฏ๏ทฏ ๐‘“ ๐‘ฅ๏ทฏ= ๐‘“ 0๏ทฏ & ๐‘“ ๐‘ฅ๏ทฏ = โˆ’1 ๐‘“ 0๏ทฏ = โˆ’1 Thus L.H.L = R.H.L = ๐‘“ 0๏ทฏ Hence ๐’‡ ๐’™๏ทฏ is continuous at x = 0 Case 2 At x > 0 ๐‘“ ๐‘ฅ๏ทฏ = โˆ’1 it is a constant function. & Every constant function is continuous for all real number. โ‡’ ๐’‡ ๐’™๏ทฏ is continuous for x > 0 . Case 3 At x = โˆ’ 1 A function is continuous at x = โˆ’1 if L.H.L = R.H.L = ๐‘“ โˆ’1๏ทฏ & ๐‘“ ๐‘ฅ๏ทฏ = โˆ’2๐‘ฅโˆ’1 ๐‘“ โˆ’1๏ทฏ = โˆ’2 โˆ’1๏ทฏโˆ’1 = 2 โ€“ 1 = 1 Thus L.H.L = R.H.L = ๐‘“ โˆ’1๏ทฏ Hence ๐’‡ ๐’™๏ทฏ is continuous at x = โˆ’1 Case 4 At x < โˆ’1 ๐‘“ ๐‘ฅ๏ทฏ = 1 , it is a constant function & Every constant function is continuous for all real number โ‡’ ๐’‡ ๐’™๏ทฏ is continuous for x < โˆ’1 Case 5 At โˆ’1โ‰ค๐‘ฅ<0 ๐‘“ ๐‘ฅ๏ทฏ = โˆ’2๐‘ฅโˆ’1, is a polynomial & we know that every polynomial function is continuous for all real values . Hence ๐’‡ ๐’™๏ทฏ = โˆ’๐Ÿ๐’™โˆ’๐Ÿ is continuous at โˆ’๐Ÿโ‰ค๐’™<๐ŸŽ โ‡’ There is no point of discontinuity Hence ๐’‡ ๐’™๏ทฏ is continuous for all ๐’™โˆˆ๐‘น

About the Author

Davneet Singh's photo - Teacher, Computer Engineer, Marketer
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.