Advertisements
Advertisements
प्रश्न
If \[f : R \to \left( - 1, 1 \right)\] is defined by
\[f\left( x \right) = \frac{- x|x|}{1 + x^2}, \text{ then } f^{- 1} \left( x \right)\] equals
विकल्प
\[\sqrt{\frac{\left| x \right|}{1 - \left| x \right|}}\]
\[\text{ Sgn } \left( x \right) \sqrt{\frac{\left| x \right|}{1 - \left| x \right|}}\]
\[- \sqrt{\frac{x}{1 - x}}\]
None of these
उत्तर
(b) \[- Sgn \left( x \right) \sqrt{\frac{\left| x \right|}{1 - \left| x \right|}}\]
\[\text{We have}, f\left( x \right) = \frac{- x|x|}{1 + x^2} x \in \left( - 1, 1 \right)\]
\[\text{Case} - \left( I \right)\]
\[\text{When}, x < 0, \]
\[\text{Then}, \left| x \right| = - x\]
\[\text{And} f\left( x \right) > 0\]
\[\text{Now}, \]
\[f\left( x \right) = \frac{- x\left( - x \right)}{1 + x^2}\]
\[ \Rightarrow y = \frac{x^2}{1 + x^2}\]
\[ \Rightarrow \frac{y}{1} = \frac{x^2}{1 + x^2}\]
\[ \Rightarrow \frac{y + 1}{y - 1} = \frac{x^2 + 1 + x^2}{x^2 - 1 - x^2} \left[ \text { Using Componendo and dividendo } \right]\]
\[ \Rightarrow \frac{y + 1}{y - 1} = \frac{2 x^2 + 1}{- 1}\]
\[ \Rightarrow - \frac{y + 1}{y - 1} = 2 x^2 + 1\]
\[ \Rightarrow \frac{2y}{1 - y} = 2 x^2 \]
\[ \Rightarrow \frac{y}{1 - y} = x^2 \]
\[ \Rightarrow x = - \sqrt{\frac{y}{1 - y}} \left( \text{ As } x < 0 \right)\]
\[ \Rightarrow x = - \sqrt{\frac{\left| y \right|}{1 - \left| y \right|}} \]
\[ \left[ \text{ As } y > 0 \right]\]
\[\text{To find the inverse interchanging x and y we get}, \]
\[ f^{- 1} \left( x \right) = - \sqrt{\frac{\left| x \right|}{1 - \left| x \right|}} . . . \left( i \right)\]
\[\text{Case} - \left( II \right)\]
\[\text{When}, x > 0, \]
\[\text{Then}, \left| x \right| = x\]
\[\text{And} f\left( x \right) < 0\]
\[\text{Now}, \]
\[f\left( x \right) = \frac{- x\left( x \right)}{1 + x^2}\]
\[ \Rightarrow y = \frac{- x^2}{1 + x^2}\]
\[ \Rightarrow \frac{y}{1} = \frac{- x^2}{1 + x^2}\]
\[ \Rightarrow \frac{y + 1}{y - 1} = \frac{- x^2 + 1 + x^2}{- x^2 - 1 - x^2} \left[ \text{Using Componendo and dividendo} \right]\]
\[ \Rightarrow \frac{y + 1}{y - 1} = \frac{1}{- 2 x^2 - 1}\]
\[ \Rightarrow \frac{1 + y}{1 - y} = \frac{1}{2 x^2 + 1}\]
\[ \Rightarrow \frac{1 - y}{1 + y} = 2 x^2 + 1\]
\[ \Rightarrow \frac{- 2y}{1 + y} = 2 x^2 \]
\[ \Rightarrow x^2 = \frac{- y}{1 + y}\]
\[ \Rightarrow x = \sqrt{\frac{- y}{1 + y}} \left( \text{As} x > 0 \right)\]
\[ \Rightarrow x = \sqrt{\frac{\left| y \right|}{1 - \left| y \right|}} \]
\[ \left[ \text{ As } y < 0 \right]\]
\[\text{To find the inverse interchanging x and y we get}, \]
\[ f^{- 1} \left( x \right) = \sqrt{\frac{\left| x \right|}{1 - \left| x \right|}} . . . \left( ii \right)\]
\[\text{Case} - \left( III \right)\]
\[\text{When}, x = 0, \]
\[\text{Then}, f\left( x \right) = 0\]
\[\text{Hence}, f^{- 1} \left( x \right) = 0 . . . \left( iii \right)\]
\[\text{Combinig equation} \left( i \right) , \left( ii \right) \text{and} \left( iii \right) \text{we get}, \]
\[ f^{- 1} \left( x \right) = - Sgn\left( x \right)\sqrt{\frac{\left| x \right|}{1 - \left| x \right|}}\]
APPEARS IN
संबंधित प्रश्न
Show that the function f in `A=R-{2/3} ` defined as `f(x)=(4x+3)/(6x-4)` is one-one and onto hence find f-1
Let f: R → R be defined as f(x) = 3x. Choose the correct answer.
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = x3 + 1
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = sin2x + cos2x
Let A = [-1, 1]. Then, discuss whether the following function from A to itself is one-one, onto or bijective : `f (x) = x/2`
Let A = [-1, 1]. Then, discuss whether the following functions from A to itself is one-one, onto or bijective : h(x) = x2
Find gof and fog when f : R → R and g : R → R is defined by f(x) = 2x + x2 and g(x) = x3
Let A = {a, b, c}, B = {u v, w} and let f and g be two functions from A to B and from B to A, respectively, defined as :
f = {(a, v), (b, u), (c, w)}, g = {(u, b), (v, a), (w, c)}.
Show that f and g both are bijections and 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.
Consider f : N → N, g : N → N and h : N → R defined as f(x) = 2x, g(y) = 3y + 4 and h(z) = sin z for all x, y, z ∈ N. Show that ho (gof) = (hog) of.
If f(x) = |x|, prove that fof = f.
If f(x) = sin x and g(x) = 2x be two real functions, then describe gof and fog. Are these equal functions?
Let f be any real function and let g be a function given by g(x) = 2x. Prove that gof = f + f.
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 .`
If f : A → A, g : A → A are two bijections, then prove that fog is a surjection ?
If f : C → C is defined by f(x) = x2, write f−1 (−4). Here, C denotes the set of all complex numbers.
Let \[f : \left[ - \frac{\pi}{2}, \frac{\pi}{2} \right] \to\] A be defined by f(x) = sin x. If f is a bijection, write set A.
Let f : R − {−1} → R − {1} be given by\[f\left( x \right) = \frac{x}{x + 1} . \text{Write } f^{- 1} \left( x \right)\]
Write the domain of the real function f defined by f(x) = `sqrt (25 -x^2)` [NCERT EXEMPLAR]
Let f, g : R → R be defined by f(x) = 2x + l and g(x) = x2−2 for all x
∈ R, respectively. Then, find gof. [NCERT EXEMPLAR]
If a function g = {(1, 1), (2, 3), (3, 5), (4, 7)} is described by g(x) = \[\alpha x + \beta\] then find the values of \[\alpha\] and \[ \beta\] . [NCERT EXEMPLAR]
Let\[A = \left\{ x \in R : - 1 \leq x \leq 1 \right\} = \text{B and C} = \left\{ x \in R : x \geq 0 \right\} and\]\[S = \left\{ \left( x, y \right) \in A \times B : x^2 + y^2 = 1 \right\} \text{and } S_0 = \left\{ \left( x, y \right) \in A \times C : x^2 + y^2 = 1 \right\}\]
Then,
A function f from the set of natural numbers to the set of integers defined by
\[f\left( n \right)\begin{cases}\frac{n - 1}{2}, & \text{when n is odd} \\ - \frac{n}{2}, & \text{when n is even}\end{cases}\]
Which of the following functions from
\[A = \left\{ x \in R : - 1 \leq x \leq 1 \right\}\]
Mark the correct alternative in the following question:
If the set A contains 5 elements and the set B contains 6 elements, then the number of one-one and onto mappings from A to B is
Let A = R − (2) and B = R − (1). If f: A ⟶ B is a function defined by`"f(x)"=("x"-1)/("x"-2),` how that f is one-one and onto. Hence, find f−1.
Which function is used to check whether a character is alphanumeric or not?
Let f: R → R be defined by f(x) = 3x – 4. Then f–1(x) is given by ______.
Let f: R → R be defined by f(x) = x2 + 1. Then, pre-images of 17 and – 3, respectively, are ______.
Let X = {1, 2, 3}and Y = {4, 5}. Find whether the following subset of X ×Y are function from X to Y or not
h = {(1,4), (2, 5), (3, 5)}
Let f: R → R be the functions defined by f(x) = x3 + 5. Then f–1(x) is ______.
Let f: R – `{3/5}` → R be defined by f(x) = `(3x + 2)/(5x - 3)`. Then ______.
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.
- Let R: B → G be defined by R = { (b1,g1), (b2,g2),(b3,g1)}, then R is ____________.
Raji visited the Exhibition along with her family. The Exhibition had a huge swing, which attracted many children. Raji found that the swing traced the path of a Parabola as given by y = x2.
Answer the following questions using the above information.
- Let f: R → R be defined by f(x) = x2 is:
If f; R → R f(x) = 10x + 3 then f–1(x) is:
Prove that the function f is surjective, where f: N → N such that `f(n) = {{:((n + 1)/2",", if "n is odd"),(n/2",", if "n is even"):}` Is the function injective? Justify your answer.
Consider a set containing function A= {cos–1cosx, sin(sin–1x), sinx((sinx)2 – 1), etan{x}, `e^(|cosx| + |sinx|)`, sin(tan(cosx)), sin(tanx)}. B, C, D, are subsets of A, such that B contains periodic functions, C contains even functions, D contains odd functions then the value of n(B ∩ C) + n(B ∩ D) is ______ where {.} denotes the fractional part of functions)
