Advertisements
Advertisements
प्रश्न
Check the injectivity and surjectivity of the following function:
f : Z → Z given by f(x) = x2
Advertisements
उत्तर
f : Z → Z given by f(x) = x2
Here, Z = {0, ±1, ±2, ±3, ....}
Injectivity:
Suppose f(–1) = f(1) = 1
But –1 ≠ 1
∴ f is not a one-one function.
Surjective:
There are many elements in a codomain Z which have no pre-image in the domain Z.
For example, 2 ∈ Z is an element of a codomain, but for f(x) = x2 there is no x ∈ Z such that f(x) = 2, because `sqrt2` is not an integer.
∴ f is not onto.
Hence, f is neither injective nor surjective.
APPEARS IN
संबंधित प्रश्न
Let A = R – {3} and B = R – {1}. Consider the function f : A → B defined by f(x) = `((x - 2)/(x - 3))`. Is f one-one and onto? Justify your answer.
Show that the function f : R → R given by f(x) = x3 is injective.
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)
Give an example of a function which is neither one-one nor onto ?
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = |x|
Classify the following function as injection, surjection or bijection :
f : Z → Z, defined by f(x) = x − 5
Show that the exponential function f : R → R, given by f(x) = ex, is one-one but not onto. What happens if the co-domain is replaced by`R0^+` (set of all positive real numbers)?
If f : A → B and g : B → C are one-one functions, show that gof is a one-one function.
If f : A → B and g : B → C are onto functions, show that gof is a onto function.
Find fog and gof if : f (x) = |x|, g (x) = sin x .
Let f(x) = x2 + x + 1 and g(x) = sin x. Show that fog ≠ gof.
If f(x) = |x|, prove that fof = f.
State with reason whether the following functions have inverse:
h : {2, 3, 4, 5} → {7, 9, 11, 13} with h = {(2, 7), (3, 9), (4, 11), (5, 13)}
If f : Q → Q, g : Q → Q are two functions defined by f(x) = 2 x and g(x) = x + 2, show that f and g are bijective maps. Verify that (gof)−1 = f−1 og −1.
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]
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.
If f : A → A, g : A → A are two bijections, then prove that fog is an injection ?
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 f : R → R+ be defined by f(x) = ax, a > 0 and a ≠ 1. Write f−1 (x).
Write the domain of the real function
`f (x) = sqrt([x] - x) .`
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
\[A = \left\{ x : - 1 \leq x \leq 1 \right\} \text{and} f : A \to \text{A such that f}\left( x \right) = x|x|\]
The function f : [-1/2, 1/2, 1/2] → [-π /2,π/2], defined by f (x) = `sin^-1` (3x - `4x^3`), is
Let
\[f : R - \left\{ n \right\} \to R\]
The function
Which of the following functions from
\[A = \left\{ x \in R : - 1 \leq x \leq 1 \right\}\]
If \[g \left( f \left( x \right) \right) = \left| \sin x \right| \text{and} f \left( g \left( x \right) \right) = \left( \sin \sqrt{x} \right)^2 , \text{then}\]
The distinct linear functions that map [−1, 1] onto [0, 2] are
Mark the correct alternative in the following question:
Let f : R → R be given by f(x) = tanx. Then, f-1(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.
Which of the following functions from Z into Z is bijective?
Range of `"f"("x") = sqrt((1 - "cos x") sqrt ((1 - "cos x")sqrt ((1 - "cos x")....infty))`
Let R be a relation on the set L of lines defined by l1 R l2 if l1 is perpendicular to l2, then relation R 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 ____________.
Let [x] denote the greatest integer ≤ x, where x ∈ R. If the domain of the real valued function f(x) = `sqrt((|[x]| - 2)/(|[x]| - 3)` is (–∞, a) ∪ [b, c) ∪ [4, ∞), a < b < c, then the value of a + b + c is ______.
Let A = {1, 2, 3, ..., 10} and f : A `rightarrow` A be defined as
f(k) = `{{:(k + 1, if k "is odd"),( k, if k "is even"):}`.
Then the number of possible functions g : A `rightarrow` A such that gof = f is ______.
