Advertisements
Advertisements
प्रश्न
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
विकल्प
many-one and onto
many-one and into
one-one and into
one-one and onto
उत्तर
(b) many-one and into
f : R → R
\[f\left( x \right) = \left[ x^2 \right] + \left[ x + 1 \right] - 3\]
It is many one function because in this case for two different values of x
we would get the same value of f(x) .
\[For \]
\[x = 1 . 1, 1 . 2 \in R\]
\[f(1 . 1) = \left[ \left( 1 . 1 \right)^2 \right] + \left[ 1 . 1 + 1 \right] - 3\]
\[ = \left[ 1 . 21 \right] + \left[ 2 . 1 \right] - 3\]
\[ = 1 + 2 - 3\]
\[ = 0\]
\[f(1 . 1) = \left[ \left( 1 . 2 \right)^2 \right] + \left[ 1 . 2 + 1 \right] - 3\]
\[ = \left[ 1 . 44 \right] + \left[ 2 . 2 \right] - 3\]
\[ = 1 + 2 - 3\]
\[ = 0\]
It is into function because for the given domain we would only get the integral values of
f(x).
but R is the codomain of the given function.
That means , Codomain\[\neq\]Range Hence, the given function is into function.
Therefore, f(x) is many one and into
APPEARS IN
संबंधित प्रश्न
Let f: N → N be defined by f(n) = `{((n+1)/2, ",if n is odd"),(n/2,",n is even"):}` for all n ∈ N.
State whether the function f is bijective. Justify your answer.
Show that function f: R `rightarrow` {x ∈ R : −1 < x < 1} defined by f(x) = `x/(1 + |x|)`, x ∈ R is one-one and onto function.
Let A = [-1, 1]. Then, discuss whether the following function from A to itself is one-one, onto or bijective : `f (x) = x/2`
Show that if f1 and f2 are one-one maps from R to R, then the product f1 × f2 : R → R defined by (f1 × f2) (x) = f1 (x) f2 (x) need not be one - one.
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.
Find fog and gof if : f (x) = ex g(x) = loge x .
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?
if f (x) = `sqrt (x +3) and g (x) = x ^2 + 1` be two real functions, then find fog and gof.
Let
f (x) =`{ (1 + x, 0≤ x ≤ 2) , (3 -x , 2 < x ≤ 3):}`
Find fof.
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
If A = {1, 2, 3, 4} and B = {a, b, c, d}, define any four bijections from A to B. Also give their inverse functions.
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 : C → C is defined by f(x) = (x − 2)3, write f−1 (−1).
Let A = {1, 2, 3, 4} and B = {a, b} be two sets. Write the total number of onto functions from A to B.
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]
Write the domain of the real function f defined by f(x) = `sqrt (25 -x^2)` [NCERT EXEMPLAR]
\[f : A \to \text{B given by } 3^{ f\left( x \right)} + 2^{- x} = 4\] is a bijection, then
Which of the following functions form Z to itself are bijections?
Let [x] denote the greatest integer less than or equal to x. If \[f\left( x \right) = \sin^{- 1} x, g\left( x \right) = \left[ x^2 \right]\text{ and } h\left( x \right) = 2x, \frac{1}{2} \leq x \leq \frac{1}{\sqrt{2}}\]
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.
Let f, g: R → R be two functions defined as f(x) = |x| + x and g(x) = x – x ∀ x ∈ R. Then, find f o g and g o f
Let R be the set of real numbers and f: R → R be the function defined by f(x) = 4x + 5. Show that f is invertible and find 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.
Are the following set of ordered pairs functions? If so, examine whether the mapping is injective or surjective.
{(a, b): a is a person, b is an ancestor of a}
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
g = {(1, 4), (2, 4), (3, 4)}
Let f : R `->` R be a function defined by f(x) = x3 + 4, then f is ______.
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?
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: {1,2,3,....} → {1,4,9,....} be defined by f(x) = x2 is ____________.
A function f: x → y is said to be one – one (or injective) if:
Function f: R → R, defined by f(x) = `x/(x^2 + 1)` ∀ x ∈ R is not
The domain of the function `cos^-1((2sin^-1(1/(4x^2-1)))/π)` is ______.
Let f: R→R be defined as f(x) = 2x – 1 and g: R – {1}→R be defined as g(x) = `(x - 1/2)/(x - 1)`. Then the composition function f (g(x)) is ______.
The solution set of the inequation log1/3(x2 + x + 1) + 1 > 0 is ______.
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)
The domain of function is f(x) = `sqrt(-log_0.3(x - 1))/sqrt(x^2 + 2x + 8)` 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 ______.
Write the domain and range (principle value branch) of the following functions:
f(x) = tan–1 x.
