Advertisements
Advertisements
प्रश्न
Check the injectivity and surjectivity of the following function:
f : N → N given by f(x) = x3
Advertisements
उत्तर
f : N → N given by f(x) = x3
It is seen that for x, y ∈ N, f(x) = f(y)
⇒ x3 = y3
⇒ x = y
∴ f is injective.
Now, 2 ∈ N, but there does not exist any element x ∈ N such that f(x) = x3 = 2.
∴ f is not surjective.
Hence, function f is injective but not surjective.
APPEARS IN
संबंधित प्रश्न
In the following case, state whether the function is one-one, onto or bijective. Justify your answer.
f : R → R defined by f(x) = 1 + x2
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.
Show that the function f : R → R given by f(x) = x3 is injective.
If the function `f(x) = sqrt(2x - 3)` is invertible then find its inverse. Hence prove that `(fof^(-1))(x) = x`
Give an example of a function which is not one-one but onto ?
Give an example of a function which is neither one-one nor onto ?
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 : Z → Z, defined by f(x) = x − 5
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = sinx
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = x3 + 1
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = sin2x + cos2x
Find fog and gof if : f (x) = |x|, g (x) = sin x .
Find fog and gof if : f (x) = x+1, g(x) = `e^x`
.
If f(x) = sin x and g(x) = 2x be two real functions, then describe gof and fog. Are these equal functions?
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.
Show that the function f : Q → Q, defined by f(x) = 3x + 5, is invertible. Also, find f−1
If f : A → A, g : A → A are two bijections, then prove that fog is an injection ?
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) = x2, write f−1 (−4). Here, C denotes the set of all complex numbers.
Let f be a function from C (set of all complex numbers) to itself given by f(x) = x3. Write f−1 (−1).
If f : R → R is defined by f(x) = 3x + 2, find f (f (x)).
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
\[f\left( x \right) =^{7 - x} P_{x - 3}\]
Let
\[f : R - \left\{ n \right\} \to R\]
Let \[f\left( x \right) = x^2 and g\left( x \right) = 2^x\] Then, the solution set of the equation
Let
\[f : [2, \infty ) \to X\] be defined by
\[f\left( x \right) = 4x - x^2\] Then, f is invertible if X =
If \[g\left( x \right) = x^2 + x - 2\text{ and} \frac{1}{2} gof\left( x \right) = 2 x^2 - 5x + 2\] is equal to
Let \[f\left(x\right) = x^3\] be a function with domain {0, 1, 2, 3}. Then domain of \[f^{-1}\] is ______.
Let
\[f : R \to R\] be given by \[f\left( x \right) = x^2 - 3\] Then, \[f^{- 1}\] is given by
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
Are the following set of ordered pairs functions? If so, examine whether the mapping is injective or surjective.
{(a, b): a is a person, b is an ancestor of a}
An organization conducted a bike race under 2 different categories-boys and girls. Totally there were 250 participants. Among all of them finally, three from Category 1 and two from Category 2 were selected for the final race. Ravi forms two sets B and G with these participants for his college project. Let B = {b1,b2,b3} G={g1,g2} where B represents the set of boys selected and G the set of girls who were selected for the final race.
Ravi decides to explore these sets for various types of relations and functions.
- Ravi wants to know among those relations, how many functions can be formed from B to G?
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: N → N be defined by f(x) = x2 is ____________.
If f; R → R f(x) = 10x + 3 then f–1(x) is:
The solution set of the inequation log1/3(x2 + x + 1) + 1 > 0 is ______.
Let S = {1, 2, 3, 4, 5, 6, 7}. Then the number of possible functions f: S `rightarrow` S such that f(m.n) = f(m).f(n) for every m, n ∈ S and m.n ∈ S is equal to ______.
The function f(x) = [x], where [x] denotes the greatest integer less than or equal to x; is continuous at ______.
A perfect one-to-one matching between users and unique IDs is similar to:
