Advertisements
Advertisements
Question
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
Options
many-one and onto
many-one and into
one-one and into
one-one and onto
Advertisements
Solution
(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
RELATED QUESTIONS
Check the injectivity and surjectivity of the following function:
f : N → N given by f(x) = x2
Check the injectivity and surjectivity of the following function:
f : R → R given by f(x) = x2
Let f: R → R be the Signum Function defined as
f(x) = `{(1,x>0), (0, x =0),(-1, x< 0):}`
and g: R → R be the Greatest Integer Function given by g(x) = [x], where [x] is greatest integer less than or equal to x. Then does fog and gof coincide in (0, 1]?
Show that the function f: ℝ → ℝ defined by f(x) = `x/(x^2 + 1), ∀x in R`is neither one-one nor onto. Also, if g: ℝ → ℝ is defined as g(x) = 2x - 1. Find fog(x)
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.
Given A = {2, 3, 4}, B = {2, 5, 6, 7}. Construct an example of each of the following:
(i) an injective map from A to B
(ii) a mapping from A to B which is not injective
(iii) a mapping from A to B.
Give examples of two functions f : N → Z and g : Z → Z, such that gof is injective but gis not injective.
Find fog and gof if : f(x) = `x^2` + 2 , g (x) = 1 − `1/ (1-x)`.
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
Show that the function f : Q → Q, defined by f(x) = 3x + 5, is invertible. Also, find f−1
Which of the following graphs represents a one-one function?
If A = {1, 2, 3} and B = {a, b}, write the total number of functions from A to B.
If f : C → C is defined by f(x) = (x − 2)3, write f−1 (−1).
If f : R → R, g : R → are given by f(x) = (x + 1)2 and g(x) = x2 + 1, then write the value of fog (−3).
Let f : R → R+ be defined by f(x) = ax, a > 0 and a ≠ 1. Write f−1 (x).
Let `f : R - {- 3/5}` → R be a function defined as `f (x) = (2x)/(5x +3).`
f-1 : Range of f → `R -{-3/5}`.
Let f be an invertible real function. Write ( f-1 of ) (1) + ( f-1 of ) (2) +..... +( f-1 of ) (100 )
Write the domain of the real function
`f (x) = sqrtx - [x] .`
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 : A → B defined by
f (x) = - x2 + 6x - 8 is a bijection if
Let
\[A = \left\{ x \in R : - 1 \leq x \leq 1 \right\} = B\] Then, the mapping\[f : A \to \text{B given by} f\left( x \right) = x\left| x \right|\] is
The function f : [-1/2, 1/2, 1/2] → [-π /2,π/2], defined by f (x) = `sin^-1` (3x - `4x^3`), is
\[f : R \to R\] is defined by
\[f\left( x \right) = \frac{e^{x^2} - e^{- x^2}}{e^{x^2 + e^{- x^2}}} is\]
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)
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.
Write about strcmp() function.
Let the function f: R → R be defined by f(x) = 4x – 1, ∀ x ∈ R. Then, show that f is one-one.
The smallest integer function f(x) = [x] is ____________.
Let f : R → R be defind by f(x) = `1/"x" AA "x" in "R".` Then f is ____________.
Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. Based on the given information, f is best defined as:
If f; R → R f(x) = 10x + 3 then f–1(x) 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)
Let f(n) = `[1/3 + (3n)/100]n`, where [n] denotes the greatest integer less than or equal to n. Then `sum_(n = 1)^56f(n)` is equal to ______.
If A = {x ∈ R: |x – 2| > 1}, B = `{x ∈ R : sqrt(x^2 - 3) > 1}`, C = {x ∈ R : |x – 4| ≥ 2} and Z is the set of all integers, then the number of subsets of the set (A ∩ B ∩ C) C ∩ Z is ______.
Let a function `f: N rightarrow N` be defined by
f(n) = `{:[(2n",", n = 2"," 4"," 6"," 8","......),(n - 1",", n = 3"," 7"," 11"," 15","......),((n + 1)/2",", n = 1"," 5"," 9"," 13","......):}`
then f is ______.
The trigonometric equation tan–1x = 3tan–1 a has solution for ______.
