web analytics

Ex 1.3, 6 - Show f(x) = x/x+2 is one-one. Find inverse of f. - Finding Inverse

Slide16.JPG
Slide17.JPG

  1. Chapter 1 Class 12 Relation and Functions
  2. Serial order wise
Ask Download

Transcript

Ex1.3 , 6 Show that f: [−1, 1] → R, given by f(x) = ﷐𝑥﷮𝑥 + 2﷯ is one-one. Find the inverse of the function f: [−1, 1] → Range f. (Hint: For y ∈ Range f, y = f(x) = ﷐𝑥﷮𝑥 + 2﷯ , for some x in [−1, 1], i.e., x = ﷐2𝑦﷮1 − 𝑦﷯ ) f(x) = ﷐x﷮x+2﷯ Check one-one f(x1) = ﷐𝑥1﷮𝑥1 + 2﷯ f(x2) = ﷐𝑥2﷮𝑥2 + 2﷯ Putting f(x1) = f(x2) ﷐𝑥1﷮𝑥1 + 2﷯ = ﷐𝑥2﷮𝑥2 + 2﷯ x1(x2 + 2) = x2(x1 + 2) x1x2 + 2x1 = x2x1 + 2x2 x1x2 – x2x1 + 2x1 = 2x2 0 + 2x1 = 2x2 2x1 = 2x2 ⇒ x1 = x2 Hence, if f(x1) = f(x2) , then x1 = x2 ∴ f is one-one Finding inverse f(x) = ﷐𝑥﷮𝑥 + 2﷯ Putting f(x) = y y = ﷐𝑥﷮𝑥 + 2﷯ y(x + 2) = x yx + 2y = x yx – x = –2y x(y – 1) = –2y x = ﷐−2𝑦 ﷮𝑦 −1﷯ x = ﷐−2𝑦 ﷮−1﷐−𝑦 + 1﷯﷯ x = ﷐2𝑦 ﷮﷐1 − 𝑦﷯﷯ Let g(y) = ﷐2𝑦 ﷮﷐1 − 𝑦﷯﷯ where g: R → [−1, 1] Thus, g is inverse of f Inverse of f = g(y) = ﷐𝟐𝒚 ﷮﷐𝟏 − 𝒚﷯﷯

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 provides courses for Mathematics from Class 9 to 12. You can ask questions here.
Jail