Advertisements
Advertisements
प्रश्न
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = x3 − x
Advertisements
उत्तर
f : R → R, defined by f(x) = x3 − 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)
x3−x=y3−y
Here, we cannot say x=y.
For example, x=1 and y=-1
x3−x =1−1= 0
y3−y=(−1)3−(−1)−1+1=0
So, 1 and -1 have the same image 0 .
So, f is not an injection.
Surjection test :
Let y be any element in the co-domain (R), such that f(x) = y for some element x in R (domain).
f(x) = y
x3 − x = y
By observation we can say that there exist some x in R, such that x3 - x = y.
So, f is a surjection and f is not a bijection.
APPEARS IN
संबंधित प्रश्न
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, 3), (b, 2), (c, 1)}
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)}
Which of the following functions from A to B are one-one and onto?
f2 = {(2, a), (3, b), (4, c)} ; A = {2, 3, 4}, B = {a, b, c}
Classify the following function as injection, surjection or bijection : f : Z → Z given by f(x) = x3
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = `x/(x^2 +1)`
Show that the function f : R − {3} → R − {2} given by f(x) = `(x-2)/(x-3)` is a bijection.
Let A = {1, 2, 3}. Write all one-one from A to itself.
Show that the logarithmic function f : R0+ → R given by f (x) loga x ,a> 0 is a bijection.
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) = 2x + x2 and g(x) = x3
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.
If f : A → B and g : B → C are one-one functions, show that gof is a one-one function.
Find fog and gof if : f (x) = x+1, g (x) = sin x .
Let
f (x) =`{ (1 + x, 0≤ x ≤ 2) , (3 -x , 2 < x ≤ 3):}`
Find fof.
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).
Which one of the following graphs represents a function?
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.
Let C denote the set of all complex numbers. A function f : C → C is defined by f(x) = x3. Write f−1(1).
If f : R → R is defined by f(x) = x2, find f−1 (−25).
If f : {5, 6} → {2, 3} and g : {2, 3} → {5, 6} are given by f = {(5, 2), (6, 3)} and g = {(2, 5), (3, 6)}, then find fog. [NCERT EXEMPLAR]
Let
\[f : R \to R\] be a function defined by
Let
\[f : R \to R\] is defined by
\[f\left( x \right) = \frac{e^{x^2} - e^{- x^2}}{e^{x^2 + e^{- x^2}}} is\]
If \[f\left( x \right) = \sin^2 x\] and the composite function \[g\left( f\left( x \right) \right) = \left| \sin x \right|\] then g(x) is equal to
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)
A function f: R→ R defined by f(x) = `(3x) /5 + 2`, x ∈ R. Show that f is one-one and onto. Hence find f−1.
Which function is used to check whether a character is alphanumeric or not?
Write about strcmp() function.
Show that the function f: R → R defined by f(x) = `x/(x^2 + 1)`, ∀ ∈ + R , is neither one-one nor onto
Let D be the domain of the real valued function f defined by f(x) = `sqrt(25 - x^2)`. Then, write D
If the set A contains 5 elements and the set B contains 6 elements, then the number of one-one and onto mappings from A to B is ______.
The function f : A → B defined by f(x) = 4x + 7, x ∈ R is ____________.
The smallest integer function f(x) = [x] is ____________.
The number of bijective functions from set A to itself when A contains 106 elements is ____________.
If f; R → R f(x) = 10x + 3 then f–1(x) is:
Prove that the function f is surjective, where f: N → N such that `f(n) = {{:((n + 1)/2",", if "n is odd"),(n/2",", if "n is even"):}` Is the function injective? Justify your answer.
The domain of the function `cos^-1((2sin^-1(1/(4x^2-1)))/π)` is ______.
Let f: R→R be a continuous function such that f(x) + f(x + 1) = 2, for all x ∈ R. If I1 = `int_0^8f(x)dx` and I2 = `int_(-1)^3f(x)dx`, then the value of I1 + 2I2 is equal to ______.
