Advertisements
Advertisements
Question
If f : R → (0, 2) defined by `f (x) =(e^x - e^(x))/(e^x +e^(-x))+1`is invertible , find f-1.
Advertisements
Solution
Injectivity of f :
Let x and y be two elements of domain (R), such that
f (x) = f (y)
⇒ `(e^x - e^(-x))/(e^x -e^(-x)) +1 =(e^y - e^(-y))/(e^y -e^(-y)) + 1`
⇒`(e^x - e^(-x))/(e^x -e^(-x))= (e^y - e^(-y))/(e^y -e^(-y))`
⇒ `(e^-x(e^(2x) -1))/(e^-x(e^(2x)+1)) = (e^-y(e^(2y) -1))/(e^-y(e^(2y)+1)) `
⇒ `(e^(2x) -1)/(e^(2x) +1) = (e^(2y) -1)/(e^(2y) +1)`
⇒ (e2x−1) (e2y+1) = (e2x+1) (e2y−1)
⇒ e2x+2y + e2x−e2y −1= e2x+2y − e2x + e2y − 1
⇒ 2 × e2x =2 × e2y
⇒ e2x = e2y
⇒ 2x = 2y
⇒ x = y
So, f is one-one.
Surjectivity of f:
Let y be in the co-domain (0,2) such that f(x) = y.
`(e^x - e^-x)/(e^x +e^-x) + 1 = y `
⇒ `(e^-x(e^(2x) -1))/(e^-x(e^(2x)+1))+1 = y`
⇒ `(e^-x(e^(2x) -1))/(e^-x(e^(2x)+1)) = y - 1`
⇒ `e^(2x) -1 = (y - 1) (e^(2y) + 1)`
⇒ `e^(2x) -1 = y xx e^(2x) +y - e^(2x) -1`
⇒ `e^(2x) = y xx e^(2x) + y -e^(2x)`
⇒ `e^(2x) (2- y) = y`
⇒ `e^(2x) = y/(2-y)`
⇒ `2x = log_e (y/(2-y))`
⇒ `x = 1/2 log_e (y/(2 -y)) in R` (domain)
So, f is onto.
∴ f is a bijection and, hence, it is invertible.
Finding f -1:
Let f−1 (x) = y ...(1)
⇒ f (y) = x
⇒ `(e^y - e^-y)/(e^y + e^-y )+ 1 = x`
⇒ `(e^-y(e^(2y) -1))/(e^-y(e^(2y)+1)) + 1 = x`
⇒ `(e^-y(e^(2y) -1))/(e^-y(e^(2y)+1)) = x -1`
⇒ e2y −1 = ( x −1) ( e2y + 1 )
⇒ e2y − 1 = x × e2y + x − e2y − 1
⇒ e2y = x × e2y+ x − e2y
⇒ e2y ( 2 − x ) = x
⇒ `e^(2y) = x/(2-x)`
⇒`2y = log_e (x/(2-x))`
⇒`y =1/2 log_e (x/(2-x)) in R` (domain)
⇒`y =1/2 log_e (x/(2-x)) = f^-1 (x)` [from (1)]
` So, f^-1 (x) = 1/2 log_e (x/(2-x))`
APPEARS IN
RELATED QUESTIONS
In the following case, state whether the function is one-one, onto or bijective. Justify your answer.
f : R → R defined by f(x) = 3 − 4x
Let A = R − {3} and B = R − {1}. Consider the function f : A → B defined by f(x) = `((x- 2)/(x -3))`. Is f one-one and onto? Justify your answer.
Let f: R → R be defined as f(x) = 10x + 7. Find the function g: R → R such that g o f = f o g = 1R.
Give examples of two functions f: N → Z and g: Z → Z such that g o f is injective but gis not injective.
(Hint: Consider f(x) = x and g(x) =|x|)
Which of the following functions from A to B are one-one and onto?
f2 = {(2, a), (3, b), (4, c)} ; A = {2, 3, 4}, B = {a, b, c}
Let A = {−1, 0, 1} and f = {(x, x2) : x ∈ A}. Show that f : A → A is neither one-one nor onto.
Classify the following function as injection, surjection or bijection : f : N → N given by f(x) = x3
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = sin2x + cos2x
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = 3 − 4x
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = `x/(x^2 +1)`
Let A = [-1, 1]. Then, discuss whether the following function from A to itself is one-one, onto or bijective : g(x) = |x|
Set of ordered pair of a function ? If so, examine whether the mapping is injective or surjective :{(a, b) : a is a person, b is an ancestor of a}
If A = {1, 2, 3}, show that a onto function f : A → A must be one-one.
If f : A → B and g : B → C are onto functions, show that gof is a onto function.
If f, g : R → R be two functions defined as f(x) = |x| + x and g(x) = |x|- x, ∀x∈R" .Then find fog and gof. Hence find fog(–3), fog(5) and gof (–2).
Find f −1 if it exists : f : A → B, where A = {1, 3, 5, 7, 9}; B = {0, 1, 9, 25, 49, 81} and f(x) = x2
Consider f : R+ → [−5, ∞) given by f(x) = 9x2 + 6x − 5. Show that f is invertible with `f^-1 (x) = (sqrt (x +6)-1)/3 .`
Let A and B be two sets, each with a finite number of elements. Assume that there is an injective map from A to B and that there is an injective map from B to A. Prove that there is a bijection from A to B.
If f : R → R is defined by f(x) = 10 x − 7, then write f−1 (x).
If f(x) = 4 −( x - 7)3 then write f-1 (x).
The function
\[f : R \to R\] defined by\[f\left( x \right) = \left( x - 1 \right) \left( x - 2 \right) \left( x - 3 \right)\]
(a) one-one but not onto
(b) onto but not one-one
(c) both one and onto
(d) neither one-one nor onto
If \[f : R \to R\] is given by \[f\left( x \right) = x^3 + 3, \text{then} f^{- 1} \left( x \right)\] is equal to
Let the function f: R → R be defined by f(x) = 4x – 1, ∀ x ∈ R. Then, show that f is one-one.
Let f: R → R be the function defined by f(x) = 4x – 3 ∀ x ∈ R. Then write f–1
For sets A, B and C, let f: A → B, g: B → C be functions such that g o f is injective. Then both f and g are injective functions.
Let the function f: R → R be defined by f(x) = cosx, ∀ x ∈ R. Show that f is neither one-one nor onto
Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:
g(x) = |x|
Let A = {1, 2, 3, ...n} and B = {a, b}. Then the number of surjections from A into B is ______.
An organization conducted a bike race under 2 different categories-boys and girls. Totally there were 250 participants. Among all of them finally, three from Category 1 and two from Category 2 were selected for the final race. Ravi forms two sets B and G with these participants for his college project. Let B = {b1,b2,b3} G={g1,g2} where B represents the set of boys selected and G the set of girls who were selected for the final race.
Ravi decides to explore these sets for various types of relations and functions.
- Ravi wants to find the number of injective functions from B to G. How many numbers of injective functions are possible?
Let the function f: R → R be defined by f(x) = 4x – 1, ∀ x ∈ R then 'f' is
Consider a function f: `[0, pi/2] ->` R, given by f(x) = sinx and `g[0, pi/2] ->` R given by g(x) = cosx then f and g are
If log102 = 0.3010.log103 = 0.4771 then the number of ciphers after decimal before a significant figure comes in `(5/3)^-100` is ______.
The solution set of the inequation log1/3(x2 + x + 1) + 1 > 0 is ______.
Let f(x) be a polynomial of degree 3 such that f(k) = `-2/k` for k = 2, 3, 4, 5. Then the value of 52 – 10f(10) is equal to ______.
Find the domain of sin–1 (x2 – 4).
Which one of the following graphs is a function of x?
 |
 |
| Graph A | Graph B |
The trigonometric equation tan–1x = 3tan–1 a has solution for ______.
