Advertisements
Advertisements
प्रश्न
\[f : R \to R\] is defined by
\[f\left( x \right) = \frac{e^{x^2} - e^{- x^2}}{e^{x^2 + e^{- x^2}}} is\]
पर्याय
one-one but not onto
many-one but onto
one-one and onto
neither one-one nor onto
Advertisements
उत्तर
(d) neither one-one nor onto
\[We have, \]
\[f\left( x \right) = \frac{e^{x^2} - e^{- x^2}}{e^{x^2 + e^{- x^2}}}\]
\[\text{Here}, - 2, 2 \in R\]
\[Now, 2 \neq - 2\]
\[\text{But}, f\left( 2 \right) = f\left( - 2 \right)\]
\[\text{Therefore, function is not one - one} . \]
\[\text{And}, \]
\[\text{The minimum value of the function is 0 and maximum value is} 1\]
\[\text{That is range of the function is} \left[ 0, 1 \right] \text{but the co - domain of the function is given } R . \]
\[\text{Therefore, function is not onto} . \]
\[ \therefore \text{function is neither one - one nor onto} . \]
APPEARS IN
संबंधित प्रश्न
Check the injectivity and surjectivity of the following function:
f : Z → Z given by f(x) = x3
Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. Show that f is one-one.
Give examples of two functions f: N → Z and g: Z → Z such that g o f is injective but gis not injective.
(Hint: Consider f(x) = x and g(x) =|x|)
Give an example of a function which is neither one-one nor onto ?
Let A = {−1, 0, 1} and f = {(x, x2) : x ∈ A}. Show that f : A → A is neither one-one nor onto.
Classify the following function as injection, surjection or bijection : f : N → N given by f(x) = x2
Classify the following function as injection, surjection or bijection : f : Z → Z given by f(x) = x2
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) = 3 − 4x
Find the number of all onto functions from the set A = {1, 2, 3, ..., n} to itself.
Show that f : R→ R, given by f(x) = x — [x], is neither one-one nor onto.
if `f (x) = sqrt(1-x)` and g(x) = `log_e` x are two real functions, then describe functions fog and gof.
Show that the function f : Q → Q, defined by f(x) = 3x + 5, is invertible. Also, find f−1
Let A = {x &epsis; R | −1 ≤ x ≤ 1} and let f : A → A, g : A → A be two functions defined by f(x) = x2 and g(x) = sin (π x/2). Show that g−1 exists but f−1 does not exist. Also, find g−1.
Let f : R − {−1} → R − {1} be given by\[f\left( x \right) = \frac{x}{x + 1} . \text{Write } f^{- 1} \left( x \right)\]
If a function\[f : [2, \infty )\text{ to B defined by f}\left( x \right) = x^2 - 4x + 5\] is a bijection, then B =
If \[g \left( f \left( x \right) \right) = \left| \sin x \right| \text{and} f \left( g \left( x \right) \right) = \left( \sin \sqrt{x} \right)^2 , \text{then}\]
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
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.
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
Set A has 3 elements and the set B has 4 elements. Then the number of injective mappings that can be defined from A to B is ______.
Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:
f(x) = `x/2`
Let A = {1, 2, 3, ...n} and B = {a, b}. Then the number of surjections from A into B is ______.
Let f: R → R be given by f(x) = tan x. Then f–1(1) is ______.
If f(x) = (4 – (x – 7)3}, then f–1(x) = ______.
Let A = {0, 1} and N be the set of natural numbers. Then the mapping f: N → A defined by f(2n – 1) = 0, f(2n) = 1, ∀ n ∈ N, is onto.
The function f : R → R defined by f(x) = 3 – 4x is ____________.
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 find the number of injective functions from B to G. How many numbers of injective functions are possible?
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.
- The function f: R → R defined by f(x) = x − 4 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: N → N be defined by f(x) = x2 is ____________.
A function f: x → y is said to be one – one (or injective) if:
Let f: R→R be defined as f(x) = 2x – 1 and g: R – {1}→R be defined as g(x) = `(x - 1/2)/(x - 1)`. Then the composition function f (g(x)) is ______.
Difference between the greatest and least value of f(x) = `(1 + (cos^-1x)/π)^2 - (1 + (sin^-1x)/π)^2` is ______.
A function f : [– 4, 4] `rightarrow` [0, 4] is given by f(x) = `sqrt(16 - x^2)`. Show that f is an onto function but not a one-one function. Further, find all possible values of 'a' for which f(a) = `sqrt(7)`.
Let f(x) be a polynomial function of degree 6 such that `d/dx (f(x))` = (x – 1)3 (x – 3)2, then
Assertion (A): f(x) has a minimum at x = 1.
Reason (R): When `d/dx (f(x)) < 0, ∀ x ∈ (a - h, a)` and `d/dx (f(x)) > 0, ∀ x ∈ (a, a + h)`; where 'h' is an infinitesimally small positive quantity, then f(x) has a minimum at x = a, provided f(x) is continuous at x = a.
Find the domain of sin–1 (x2 – 4).
The trigonometric equation tan–1x = 3tan–1 a has solution for ______.
