Advertisements
Advertisements
प्रश्न
Let
\[A = \left\{ x \in R : x \leq 1 \right\} and f : A \to A\] be defined as
\[f\left( x \right) = x \left( 2 - x \right)\] Then,
\[f^{- 1} \left( x \right)\] is
पर्याय
\[1 + \sqrt{1 - x}\]
\[1 - \sqrt{1 - x}\]
\[\sqrt{1 - x}\]
\[1 \pm \sqrt{1 - x}\]
Advertisements
उत्तर
LaTeX
\[\text{Let y be the element in the codomain R such that}\]
\[ f^{- 1} \left( x \right) = y . . . \left( 1 \right)\]
\[ \Rightarrow f\left( y \right) = x and y \leq1 \]
\[ \Rightarrow y\left( 2 - y \right) = x\]
\[ \Rightarrow 2y - y^2 = x\]
\[ \Rightarrow y^2 - 2y + x = 0\]
\[ \Rightarrow y^2 - 2y = - x\]
\[ \Rightarrow y^2 - 2y + 1 = 1 - x\]
\[ \Rightarrow \left( y - 1 \right)^2 = 1 - x\]
\[ \Rightarrow y - 1 = \pm \sqrt{1 - x}\]
\[ \Rightarrow y = 1 \pm \sqrt{1 - x}\]
\[ \Rightarrow y = 1 - \sqrt{1 - x} \left ( \because y \leq1 \right)\]
The correct answer is (b).
APPEARS IN
संबंधित प्रश्न
Prove that the function f : N → N, defined by f(x) = x2 + x + 1, is one-one but not onto
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = 1 + x2
Show that the logarithmic function f : R0+ → R given by f (x) loga x ,a> 0 is a bijection.
Find the number of all onto functions from the set A = {1, 2, 3, ..., n} to itself.
Give examples of two surjective functions f1 and f2 from Z to Z such that f1 + f2 is not surjective.
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.
Give examples of two functions f : N → N and g : N → N, such that gof is onto but f is not onto.
Find fog and gof if : f (x) = x+1, g(x) = `e^x`
.
Find fog and gof if : f(x) = c, c ∈ R, g(x) = sin `x^2`
Let f(x) = x2 + x + 1 and g(x) = sin x. Show that fog ≠ gof.
If f(x) = 2x + 5 and g(x) = x2 + 1 be two real functions, then describe each of the following functions:
(1) fog
(2) gof
(3) fof
(4) f2
Also, show that fof ≠ f2
Consider f : {1, 2, 3} → {a, b, c} and g : {a, b, c} → {apple, ball, cat} defined as f (1) = a, f (2) = b, f (3) = c, g (a) = apple, g (b) = ball and g (c) = cat. Show that f, g and gof are invertible. Find f−1, g−1 and gof−1and show that (gof)−1 = f −1o g−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)}.
If f : R → R is defined by f(x) = 10 x − 7, then write f−1 (x).
If f(x) = x + 7 and g(x) = x − 7, x ∈ R, write fog (7).
Let A = {a, b, c, d} and f : A → A be given by f = {( a,b ),( b , d ),( c , a ) , ( d , c )} write `f^-1`. [NCERT EXEMPLAR]
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
Let
\[f : R \to R\] be a function defined by
Which of the following functions from
\[A = \left\{ x \in R : - 1 \leq x \leq 1 \right\}\]
If \[g \left( f \left( x \right) \right) = \left| \sin x \right| \text{and} f \left( g \left( x \right) \right) = \left( \sin \sqrt{x} \right)^2 , \text{then}\]
If \[F : [1, \infty ) \to [2, \infty )\] is given by
\[f\left( x \right) = x + \frac{1}{x}, then f^{- 1} \left( x \right)\]
Let \[f\left(x\right) = x^3\] be a function with domain {0, 1, 2, 3}. Then domain of \[f^{-1}\] is ______.
Mark the correct alternative in the following question:
Let f : R \[-\] \[\left\{ \frac{3}{5} \right\}\] \[\to\] R be defined by f(x) = \[\frac{3x + 2}{5x - 3}\] Then,
Let A = ℝ − {3}, B = ℝ − {1}. Let f : A → B be defined by \[f\left( x \right) = \frac{x - 2}{x - 3}, \forall x \in A\] Show that f is bijective. Also, find
(i) x, if f−1(x) = 4
(ii) f−1(7)
Write about strlen() function.
Let C be the set of complex numbers. Prove that the mapping f: C → R given by f(z) = |z|, ∀ z ∈ C, 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:
k(x) = x2
The number of bijective functions from set A to itself when A contains 106 elements is ____________.
Let g(x) = x2 – 4x – 5, then ____________.
A general election of Lok Sabha is a gigantic exercise. About 911 million people were eligible to vote and voter turnout was about 67%, the highest ever
Let I be the set of all citizens of India who were eligible to exercise their voting right in the general election held in 2019. A relation ‘R’ is defined on I as follows:
R = {(V1, V2) ∶ V1, V2 ∈ I and both use their voting right in the general election - 2019}
- Three friends F1, F2, and F3 exercised their voting right in general election-2019, then which of the following is true?
Let f: R → R defined by f(x) = x4. Choose the correct answer
Let f: R → R defined by f(x) = 3x. Choose the correct answer
Let [x] denote the greatest integer ≤ x, where x ∈ R. If the domain of the real valued function f(x) = `sqrt((|[x]| - 2)/(|[x]| - 3)` is (–∞, a) ∪ [b, c) ∪ [4, ∞), a < b < c, then the value of a + b + c is ______.
Let x is a real number such that are functions involved are well defined then the value of `lim_(t→0)[max{(sin^-1 x/3 + cos^-1 x/3)^2, min(x^2 + 4x + 7)}]((sin^-1t)/t)` where [.] is greatest integer function and all other brackets are usual brackets.
Let a and b are two positive integers such that b ≠ 1. Let g(a, b) = Number of lattice points inside the quadrilateral formed by lines x = 0, y = 0, x = b and y = a. f(a, b) = `[a/b] + [(2a)/b] + ... + [((b - 1)a)/b]`, then the value of `[(g(101, 37))/(f(101, 37))]` is ______.
(Note P(x, y) is lattice point if x, y ∈ I)
(where [.] denotes greatest integer function)
Let f(x) = ax (a > 0) be written as f(x) = f1(x) + f2(x), where f1(x) is an even function and f2(x) is an odd function. Then f1(x + y) + f1(x – y) equals ______.
A function is called bijective if it is both:
Which condition represents a one-one (injective) function?
