Advertisements
Advertisements
Question
The inverse of the function
\[f : R \to \left\{ x \in R : x < 1 \right\}\] given by
\[f\left( x \right) = \frac{e^x - e^{- x}}{e^x + e^{- x}}\] is
Options
\[\frac{1}{2} \log \frac{1 + x}{1 - x}\]
\[\frac{1}{2} \log \frac{2 + x}{2 - x}\]
\[\frac{1}{2} \log \frac{1 - x}{1 + x}\]
none of these
Advertisements
Solution
\[\text{Let} f^{- 1} \left( x \right) = y . . . \left( 1 \right)\]
\[ \Rightarrow f\left( y \right) = x\]
\[ \Rightarrow \frac{e^y - e^{- y}}{e^y + e^{- y}} = x\]
\[ \Rightarrow \frac{e^{- y} \left( e^{2y} - 1 \right)}{e^{- y} \left( e^{2y} + 1 \right)} = x\]
\[ \Rightarrow \left( e^{2y} - 1 \right) = x\left( e^{2y} + 1 \right)\]
\[ \Rightarrow e^{2y} - 1 = x e^{2y} + x\]
\[ \Rightarrow e^{2y} \left( 1 - x \right) = x + 1\]
\[ \Rightarrow e^{2y} = \frac{1 + x}{1 - x}\]
\[ \Rightarrow 2y = \log_e \left( \frac{1 + x}{1 - x} \right)\]
\[ \Rightarrow y = \frac{1}{2}l {og}_e \left( \frac{1 + x}{1 - x} \right)\]
\[ \Rightarrow f^{- 1} \left( x \right) = \frac{1}{2}l {og}_e \left( \frac{1 + x}{1 - x} \right) [\text{from}\left( 1 \right)]\]
So, the answer is (a).
APPEARS IN
RELATED QUESTIONS
Check the injectivity and surjectivity of the following function:
f : N → N given by f(x) = x2
Find the number of all onto functions from the set {1, 2, 3, ..., n} to itself.
Classify the following function as injection, surjection or bijection : f : Z → Z given by f(x) = x2
Classify the following function as injection, surjection or bijection :
f : Z → Z, defined by f(x) = x − 5
Give examples of two surjective functions f1 and f2 from Z to Z such that f1 + f2 is not surjective.
Show that f : R→ R, given by f(x) = x — [x], is neither one-one nor onto.
Find gof and fog when f : R → R and g : R → R is defined by f(x) = 2x + x2 and g(x) = x3
Find gof and fog when f : R → R and g : R → R is defined by f(x) = x2 + 2x − 3 and g(x) = 3x − 4 .
Let f = {(3, 1), (9, 3), (12, 4)} and g = {(1, 3), (3, 3) (4, 9) (5, 9)}. Show that gof and fog are both defined. Also, find fog and gof.
Verify associativity for the following three mappings : f : N → Z0 (the set of non-zero integers), g : Z0 → Q and h : Q → R given by f(x) = 2x, g(x) = 1/x and h(x) = ex.
Find fog and gof if : f (x) = ex g(x) = loge x .
Find fog and gof if : f(x) = c, c ∈ R, g(x) = sin `x^2`
If f(x) = |x|, prove that fof = f.
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).
Show that the function f : Q → Q, defined by f(x) = 3x + 5, is invertible. Also, find f−1
Let f : [−1, ∞) → [−1, ∞) be given by f(x) = (x + 1)2 − 1, x ≥ −1. Show that f is invertible. Also, find the set S = {x : f(x) = f−1 (x)}.
Let f : R → R be defined as `f (x) = (2x - 3)/4.` write fo f-1 (1) .
If f : {5, 6} → {2, 3} and g : {2, 3} → {5, 6} are given by f = {(5, 2), (6, 3)} and g = {(2, 5), (3, 6)}, then find fog. [NCERT EXEMPLAR]
Which one the following relations on A = {1, 2, 3} is a function?
f = {(1, 3), (2, 3), (3, 2)}, g = {(1, 2), (1, 3), (3, 1)} [NCERT EXEMPLAR]
\[f : A \to \text{B given by } 3^{ f\left( x \right)} + 2^{- x} = 4\] is a bijection, then
The function
f : A → B defined by
f (x) = - x2 + 6x - 8 is a bijection if
Let
f : R → R be given by
\[f\left( x \right) = \left[ x^2 \right] + \left[ x + 1 \right] - 3\]
where [x] denotes the greatest integer less than or equal to x. Then, f(x) is
(d) one-one and onto
Let f be an injective map with domain {x, y, z} and range {1, 2, 3}, such that exactly one of the following statements is correct and the remaining are false.
\[f\left( x \right) = 1, f\left( y \right) \neq 1, f\left( z \right) \neq 2 .\]
The value of
\[f^{- 1} \left( 1 \right)\] is
Let \[f\left( x \right) = \frac{\alpha x}{x + 1}, x \neq - 1\] Then, for what value of α is \[f \left( f\left( x \right) \right) = x?\]
Let
\[f : R \to R\] be given by \[f\left( x \right) = x^2 - 3\] Then, \[f^{- 1}\] is given by
Mark the correct alternative in the following question:
Let f : R → R be given by f(x) = tanx. Then, f-1(1) is
Let N be the set of natural numbers and the function f: N → N be defined by f(n) = 2n + 3 ∀ n ∈ N. Then f is ______.
Using the definition, prove that the function f: A→ B is invertible if and only if f is both one-one and onto
Which of the following functions from Z into Z are bijections?
Let X = {-1, 0, 1}, Y = {0, 2} and a function f : X → Y defiend by y = 2x4, is ____________.
Let f : R → R be a function defined by f(x) `= ("e"^abs"x" - "e"^-"x")/("e"^"x" + "e"^-"x")` then f(x) is
Let A = R – {3}, B = R – {1}. Let f : A → B be defined by `"f"("x") = ("x" - 2)/("x" - 3)` Then, ____________.
Let f : R → R, g : R → R be two functions such that f(x) = 2x – 3, g(x) = x3 + 5. The function (fog)-1 (x) is equal to ____________.
Let R be a relation on the set L of lines defined by l1 R l2 if l1 is perpendicular to l2, then relation R is ____________.
'If 'f' is a linear function satisfying f[x + f(x)] = x + f(x), then f(5) can be equal to:
Difference between the greatest and least value of f(x) = `(1 + (cos^-1x)/π)^2 - (1 + (sin^-1x)/π)^2` is ______.
A function f : [– 4, 4] `rightarrow` [0, 4] is given by f(x) = `sqrt(16 - x^2)`. Show that f is an onto function but not a one-one function. Further, find all possible values of 'a' for which f(a) = `sqrt(7)`.
Let \(X=\{-1,2,5,8,11\}\), \(Y=\{4,6,8\}\), and suppose different elements \(2,5,8\) of \(X\) have the same image \(4\). What type of function is it with respect to the one-one property?
