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# 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. - CBSE (Commerce) Class 12 - Mathematics

ConceptComposition of Functions and Invertible Function

#### Question

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 in Range f, y = f(x) = x/(x +2) for some x in [-1, 1] ie x = 2y/(1-y)

#### Solution

f: [−1, 1] → R is given as f(x) = x/(x + 2)

Let f(x) = f(y).

=> x/(x + 2) = y/(y +2)

=> xy + 2x = xy + 2y

=> 2x = 2y

=> x = y

∴ f is a one-one function.

It is clear that f: [−1, 1] → Range f is onto.

∴ f: [−1, 1] → Range f is one-one and onto and therefore, the inverse of the function:

f: [−1, 1] → Range exists.

Let g: Range f → [−1, 1] be the inverse of f.

Let y be an arbitrary element of range f.

Since f: [−1, 1] → Range f is onto, we have:

=> y = x/(x + 2)

=> xy + 2y = x

=> x(1-y)= 2y

=> x = (2y)/(1-y), y !=1

g(y) = (2y)/(1-y), y != 1

Now (gof) (x) = g(f(x)) = g(x/(x+2))= (2(x/(x+2)))/(1-x/(x+2)) = (2x)/(x + 2 - x) = 2x/2 = x

(fog)(y) = f(g(y)) = f("2y"/(1-y)) = ((2y)/((1-y)))/(((2y)/(1-y)) +2) = (2y)/(2y + 2 -2y) = (2y)/2 = y

:. gof = I_(-1,1) and fog - I_"Range f"

:. f^(-1) = g

=> f^(-1) (y) = "2y"/(1 - y), y != 1`

Is there an error in this question or solution?

#### APPEARS IN

NCERT Solution for Mathematics Textbook for Class 12 (2018 to Current)
Chapter 1: Relations and Functions
Q: 6 | Page no. 18

#### Video TutorialsVIEW ALL [4]

Solution 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. Concept: Composition of Functions and Invertible Function.
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