Advertisements
Advertisements
प्रश्न
The function \[f : [0, \infty ) \to \text {R given by } f\left( x \right) = \frac{x}{x + 1} is\]
पर्याय
one-one and onto
one-one but not onto
onto but not one-one
onto but not one-one
Advertisements
उत्तर
Injectivity:
Let x and y be two elements in the domain, such that
\[f\left( x \right) = f\left( y \right)\]
\[ \Rightarrow \frac{x}{x + 1} = \frac{y}{y + 1}\]
\[ \Rightarrow xy + x = xy + y\]
\[ \Rightarrow x = y\]
So, f is one-one.
Surjectivity:
Let y be an element in the co domain R, such that
\[y = f\left( x \right)\]
\[ \Rightarrow y = \frac{x}{x + 1}\]
\[ \Rightarrow xy + y = x\]
\[ \Rightarrow x\left( y - 1 \right) = - y\]
\[ \Rightarrow x = \frac{- y}{y - 1}\]
\[\text{Range off} = R - \left\{ 1 \right\} \neq \text{ co domain } (R)\]
⇒ is not onto.
So, the answer is (b)
APPEARS IN
संबंधित प्रश्न
Prove that the greatest integer function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.
If the function `f(x) = sqrt(2x - 3)` is invertible then find its inverse. Hence prove that `(fof^(-1))(x) = x`
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 : R → R, defined by f(x) = x3 + 1
Classify the following function as injection, surjection or bijection :
f : Q − {3} → Q, defined by `f (x) = (2x +3)/(x-3)`
Classify the following function as injection, surjection or bijection :
f : Q → Q, defined by f(x) = x3 + 1
If f : A → B is an injection, such that range of f = {a}, determine the number of elements in A.
Find gof and fog when f : R → R and g : R → R is defined by f(x) = x2 + 8 and g(x) = 3x3 + 1 .
Find gof and fog when f : R → R and g : R → R is defined by f(x) = x2 + 2x − 3 and g(x) = 3x − 4 .
Let f : R → R and g : R → R be defined by f(x) = x2 and g(x) = x + 1. Show that fog ≠ gof.
Give examples of two functions f : N → N and g : N → N, such that gof is onto but f is not onto.
Find fog and gof if : f (x) = |x|, g (x) = sin x .
Let
f (x) =`{ (1 + x, 0≤ x ≤ 2) , (3 -x , 2 < x ≤ 3):}`
Find fof.
Let f : [−1, ∞) → [−1, ∞) be given by f(x) = (x + 1)2 − 1, x ≥ −1. Show that f is invertible. Also, find the set S = {x : f(x) = f−1 (x)}.
If f : R → R is given by f(x) = x3, write f−1 (1).
Let C denote the set of all complex numbers. A function f : C → C is defined by f(x) = x3. Write f−1(1).
Write the domain of the real function
`f (x) = sqrtx - [x] .`
Let M be the set of all 2 × 2 matrices with entries from the set R of real numbers. Then, the function f : M→ R defined by f(A) = |A| for every A ∈ M, is
The range of the function
\[f\left( x \right) =^{7 - x} P_{x - 3}\]
A function f from the set of natural numbers to integers defined by
`{([n-1]/2," when n is odd" is ),(-n/2,when n is even ) :}`
The function f : [-1/2, 1/2, 1/2] → [-π /2,π/2], defined by f (x) = `sin^-1` (3x - `4x^3`), is
A function f from the set of natural numbers to the set of integers defined by
\[f\left( n \right)\begin{cases}\frac{n - 1}{2}, & \text{when n is odd} \\ - \frac{n}{2}, & \text{when n is even}\end{cases}\]
The distinct linear functions that map [−1, 1] onto [0, 2] are
Let
\[f : [2, \infty ) \to X\] be defined by
\[f\left( x \right) = 4x - x^2\] Then, f is invertible if X =
If \[f : R \to R\] is given by \[f\left( x \right) = x^3 + 3, \text{then} f^{- 1} \left( x \right)\] is equal to
Let f, g: R → R be two functions defined as f(x) = |x| + x and g(x) = x – x ∀ x ∈ R. Then, find f o g and g o f
Consider the set A containing n elements. Then, the total number of injective functions from A onto itself is ______
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
g = {(1, 4), (2, 4), (3, 4)}
Using the definition, prove that the function f: A→ B is invertible if and only if f is both one-one and onto
Let f: `[2, oo)` → R be the function defined by f(x) = x2 – 4x + 5, then the range of f is ______.
The number of bijective functions from set A to itself when A contains 106 elements is ____________.
The domain of the function `"f"("x") = 1/(sqrt ({"sin x"} + {"sin" ( pi + "x")}))` where {.} denotes fractional part, is
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 ______.
If f: R→R is a function defined by f(x) = `[x - 1]cos((2x - 1)/2)π`, where [ ] denotes the greatest integer function, then f is ______.
Difference between the greatest and least value of f(x) = `(1 + (cos^-1x)/π)^2 - (1 + (sin^-1x)/π)^2` is ______.
Let f: R→R be a polynomial function satisfying f(x + y) = f(x) + f(y) + 3xy(x + y) –1 ∀ x, y ∈ R and f'(0) = 1, then `lim_(x→∞)(f(2x))/(f(x)` is equal to ______.
Let f(n) = `[1/3 + (3n)/100]n`, where [n] denotes the greatest integer less than or equal to n. Then `sum_(n = 1)^56f(n)` is equal to ______.
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.
The trigonometric equation tan–1x = 3tan–1 a has solution for ______.
