Advertisements
Advertisements
प्रश्न
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = |x|
Advertisements
उत्तर
f : R → R, defined by f(x) = |x|
Injection test:
Let x and y be any two elements in the domain (R), such that f(x) = f(y)
f(x) = f(y)
|x|=|y|
x=±y
So, f is not an injection .
Surjection test :
Let y be an element in the co-domain (R), such that f(x) = y for some element x in R (domain).
f(x) = y
|x|=y
x = ± y ∈ Z
So, f is a surjection and f is not a bijection.
APPEARS IN
संबंधित प्रश्न
Check the injectivity and surjectivity of the following function:
f : R → R given by f(x) = x2
Check the injectivity and surjectivity of the following function:
f : Z → Z given by f(x) = x3
Show that the function f : R → {x ∈ R : −1 < x < 1} defined by f(x) = `x/(1 + |x|)`, x ∈ R is one-one and onto function.
Classify the following function as injection, surjection or bijection : f : Z → Z given by f(x) = x2
Find gof and fog when f : R → R and g : R → R is defined by f(x) = 2x + x2 and g(x) = x3
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) = x2 g(x) = cos x .
Find fog and gof if : f (x) = x+1, g(x) = `e^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
if f (x) = `sqrt (x +3) and g (x) = x ^2 + 1` be two real functions, then find fog and gof.
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 A = {1, 2, 3, 4}; B = {3, 5, 7, 9}; C = {7, 23, 47, 79} and f : A → B, g : B → C be defined as f(x) = 2x + 1 and g(x) = x2 − 2. Express (gof)−1 and f−1 og−1 as the sets of ordered pairs and verify that (gof)−1 = f−1 og−1.
A function f : R → R is defined as f(x) = x3 + 4. Is it a bijection or not? In case it is a bijection, find f−1 (3).
Let A = R - {3} and B = R - {1}. Consider the function f : A → B defined by f(x) = `(x-2)/(x-3).`Show that f is one-one and onto and hence find f-1.
[CBSE 2012, 2014]
Consider the function f : R+ → [-9 , ∞ ]given by f(x) = 5x2 + 6x - 9. Prove that f is invertible with f -1 (y) = `(sqrt(54 + 5y) -3)/5` [CBSE 2015]
If f : R → R is given by f(x) = x3, write f−1 (1).
Let f be a function from C (set of all complex numbers) to itself given by f(x) = x3. Write f−1 (−1).
Let A = {x ∈ R : −4 ≤ x ≤ 4 and x ≠ 0} and f : A → R be defined by \[f\left( x \right) = \frac{\left| x \right|}{x}\]Write the range of f.
Let f : R → R+ be defined by f(x) = ax, a > 0 and a ≠ 1. Write f−1 (x).
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 of the following functions from
to itself are bijections?
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 → R be given by f(x) = tanx. Then, f-1(1) is
Let f: R → R be the function defined by f(x) = 4x – 3 ∀ x ∈ R. Then write f–1
If A = {a, b, c, d} and f = {a, b), (b, d), (c, a), (d, c)}, show that f is one-one from A onto A. Find f–1
Let f: R → R be defined by f(x) = x2 + 1. Then, pre-images of 17 and – 3, respectively, are ______.
Let the function f: R → R be defined by f(x) = cosx, ∀ x ∈ R. Show that f is neither one-one nor onto
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
k = {(1,4), (2, 5)}
Let f: R → R be defined by f(x) = `1/x` ∀ x ∈ R. Then f is ______.
The number of bijective functions from set A to itself when A contains 106 elements is ____________.
Let X = {-1, 0, 1}, Y = {0, 2} and a function f : X → Y defiend by y = 2x4, is ____________.
Students of Grade 9, planned to plant saplings along straight lines, parallel to each other to one side of the playground ensuring that they had enough play area. Let us assume that they planted one of the rows of the saplings along the line y = x − 4. Let L be the set of all lines which are parallel on the ground and R be a relation on L.
Answer the following using the above information.
- Let f: R → R be defined by f(x) = x − 4. Then the range of f(x) 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.
- 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 ____________.
Number of integral values of x satisfying the inequality `(3/4)^(6x + 10 - x^2) < 27/64` 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 = R – {2} and B = R – {1}. If f: A `→` B is a function defined by f(x) = `(x - 1)/(x - 2)` then show that f is a one-one and an onto function.
The trigonometric equation tan–1x = 3tan–1 a has solution for ______.
The function defined by \[\mathrm{f}(x)=\frac{2x+3}{3x+4},x\neq-\frac{4}{3}\] is
