Advertisements
Advertisements
Question
Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:
g(x) = |x|
Solution
Given, A = [–1, 1]
Let g(x1) = g(x2)
|x1| = |x2|
x1 = ± x2
So, g(x) is not one-one
Also g(x) = |x| ≥ 0, for all real x
Hence, the range is [0, 1], which is subset of co-domain ‘A’
So, f(x) is not onto.
Therefore, f(x) is not bijective.
APPEARS IN
RELATED QUESTIONS
Let A = {−1, 0, 1, 2}, B = {−4, −2, 0, 2} and f, g: A → B be functions defined by f(x) = x2 − x, x ∈ A and g(x) = `2|x - 1/2|- 1, x in A`. Are f and g equal?
Justify your answer. (Hint: One may note that two functions f: A → B and g: A → B such that f(a) = g(a) ∀ a ∈ A are called equal functions).
Classify the following function as injection, surjection or bijection :
f : Z → Z, defined by f(x) = x2 + x
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`
If A = {1, 2, 3}, show that a onto function f : A → A must be one-one.
Give examples of two surjective functions f1 and f2 from Z to Z such that f1 + f2 is not surjective.
Find gof and fog when f : R → R and g : R → R is defined by f(x) = x2 + 8 and g(x) = 3x3 + 1 .
Find gof and fog when f : R → R and g : R → R is defined by f(x) = 8x3 and g(x) = x1/3.
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.
Find fog and gof if : f(x) = c, c ∈ R, g(x) = sin `x^2`
Find fog and gof if : f(x) = `x^2` + 2 , g (x) = 1 − `1/ (1-x)`.
If f(x) = sin x and g(x) = 2x be two real functions, then describe gof and fog. Are these equal functions?
Find f −1 if it exists : f : A → B, where A = {0, −1, −3, 2}; B = {−9, −3, 0, 6} and f(x) = 3 x.
Consider f : R → R+ → [4, ∞) given by f(x) = x2 + 4. Show that f is invertible with inverse f−1 of f given by f−1 `(x)= sqrt (x-4)` where R+ is the set of all non-negative real numbers.
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.
Write the total number of one-one functions from set A = {1, 2, 3, 4} to set B = {a, b, c}.
If f : C → C is defined by f(x) = x2, write f−1 (−4). Here, C denotes the set of all complex numbers.
If f : R → R defined by f(x) = 3x − 4 is invertible, then write f−1 (x).
Write the domain of the real function
`f (x) = 1/(sqrt([x] - x)`.
If f : R → R is defined by f(x) = 3x + 2, find f (f (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]
Let the function
\[f : R - \left\{ - b \right\} \to R - \left\{ 1 \right\}\]
\[f\left( x \right) = \frac{x + a}{x + b}, a \neq b .\text{Then},\]
The function
Let
\[f : [2, \infty ) \to X\] be defined by
\[f\left( x \right) = 4x - x^2\] Then, f is invertible if X =
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 \[f\left(x\right) = x^3\] be a function with domain {0, 1, 2, 3}. Then domain of \[f^{-1}\] is ______.
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.
Consider the set A containing n elements. Then, the total number of injective functions from A onto itself is ______
Let f : R → R be defind by f(x) = `1/"x" AA "x" in "R".` Then f is ____________.
Let n(A) = 4 and n(B) = 6, Then the number of one – one functions from 'A' to 'B' is:
