Advertisements
Advertisements
Question
Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:
f(x) = `x/2`
Advertisements
Solution
Given, A = [–1, 1]
f: [–1, 1] → [–1, 1], f(x) = `x/2`
Let f(x1) = f(x2)
`x_1/2` = x2
So, f(x) is one-one.
Also x ∈ [–1, 1]
`x/2` = f(x) = `[-1/2, 1/2]`
Hence, the range is a subset of co-domain ‘A’
So, f(x) is not onto.
Therefore, f(x) is not bijective.
APPEARS IN
RELATED QUESTIONS
Check the injectivity and surjectivity of the following function:
f : N → N given by f(x) = x2
In the following case, state whether the function is one-one, onto or bijective. Justify your answer.
f : R → R defined by f(x) = 3 – 4x
Let A and B be sets. Show that f : A × B → B × A such that f(a, b) = (b, a) is bijective function.
Let f: R → R be defined as f(x) = 10x + 7. Find the function g: R → R such that g o f = f o g = 1R.
Let S = {a, b, c} and T = {1, 2, 3}. Find F−1 of the following functions F from S to T, if it exists.
F = {(a, 2), (b, 1), (c, 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)
Let A = {−1, 0, 1} and f = {(x, x2) : x ∈ A}. Show that f : A → A is neither one-one nor onto.
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = sinx
Let A = [-1, 1]. Then, discuss whether the following functions from A to itself is one-one, onto or bijective : h(x) = x2
Set of ordered pair of a function ? If so, examine whether the mapping is injective or surjective :{(a, b) : a is a person, b is an ancestor of a}
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) = 8x3 and g(x) = x1/3.
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.
If f(x) = sin x and g(x) = 2x be two real functions, then describe gof and fog. Are these equal functions?
State with reason whether the following functions have inverse :
f : {1, 2, 3, 4} → {10} with f = {(1, 10), (2, 10), (3, 10), (4, 10)}
State with reason whether the following functions have inverse :
g : {5, 6, 7, 8} → {1, 2, 3, 4} with g = {(5, 4), (6, 3), (7, 4), (8, 2)}
Consider f : R → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f.
Which one of the following graphs represents a function?
If A = {1, 2, 3} and B = {a, b}, write the total number of functions from A to B.
If f : R → R is given by f(x) = x3, write f−1 (1).
Let f : R → R, g : R → R be two functions defined by f(x) = x2 + x + 1 and g(x) = 1 − x2. Write fog (−2).
Write the domain of the real function f defined by f(x) = `sqrt (25 -x^2)` [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
\[A = \left\{ x \in R : - 1 \leq x \leq 1 \right\} = B\] Then, the mapping\[f : A \to \text{B given by} f\left( x \right) = x\left| x \right|\] is
The range of the function
\[f\left( x \right) =^{7 - x} P_{x - 3}\]
Which of the following functions from
\[A = \left\{ x \in R : - 1 \leq x \leq 1 \right\}\]
Write about strcmp() function.
Set A has 3 elements and the set B has 4 elements. Then the number of injective mappings that can be defined from A to B is ______.
Let A be a finite set. Then, each injective function from A into itself is not surjective.
Let D be the domain of the real valued function f defined by f(x) = `sqrt(25 - x^2)`. Then, write D
If f: R → R is defined by f(x) = x2 – 3x + 2, write f(f (x))
Let the function f: R → R be defined by f(x) = cosx, ∀ x ∈ R. Show that f is neither one-one nor onto
The mapping f : N → N is given by f(n) = 1 + n2, n ∈ N when N is the set of natural numbers 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}
- Mr. ’X’ and his wife ‘W’ both exercised their voting right in the general election-2019, 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 ____________.
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.
- The function f: Z → Z defined by f(x) = x2 is ____________.
Difference between the greatest and least value of f(x) = `(1 + (cos^-1x)/π)^2 - (1 + (sin^-1x)/π)^2` is ______.
Write the domain and range (principle value branch) of the following functions:
f(x) = tan–1 x.
A function is called one-one if different elements of the domain have:
