Advertisements
Advertisements
प्रश्न
If the function\[f : R \to \text{A given by} f\left( x \right) = \frac{x^2}{x^2 + 1}\] is a surjection, then A =
पर्याय
R
[0, 1]
[0, 1]
[0, 1]
Advertisements
उत्तर
\[\text{As f is surjective, range of f = co - domain of}f\]
\[ \Rightarrow A = \text{range of f} \]
\[ \because f\left( x \right) = \frac{x^2}{x^2 + 1}, \]
\[ y = \frac{x^2}{x^2 + 1}\]
\[ \Rightarrow y\left( x^2 + 1 \right) = x^2 \]
\[ \Rightarrow \left( y - 1 \right) x^2 + y = 0\]
\[ \Rightarrow x^2 = \frac{- y}{\left( y - 1 \right)}\]
\[ \Rightarrow x = \sqrt{\frac{y}{\left( 1 - y \right)}}\]
\[ \Rightarrow \frac{y}{\left( 1 - y \right)} \geq 0\]
\[ \Rightarrow y \in [0, 1)\]
\[ \Rightarrow \text{Range of f} = [0, 1)\]
\[ \Rightarrow A = [0, 1)\]
So, the answer is (d) .
APPEARS IN
संबंधित प्रश्न
Let f : N → N be defined by f(n) = `{((n+1)/2", if n is odd"),(n/2", if n is even"):}` for all n ∈ N.
State whether the function f is bijective. Justify your answer.
Let f : R → R be defined as f(x) = 3x. Choose the correct answer.
Given examples of two functions f: N → N and g: N → N such that gof is onto but f is not onto.
(Hint: Consider f(x) = x + 1 and `g(x) = {(x-1, ifx >1),(1, if x = 1):}`
If the function `f(x) = sqrt(2x - 3)` is invertible then find its inverse. Hence prove that `(fof^(-1))(x) = x`
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) = 5x3 + 4
Give examples of two surjective functions f1 and f2 from Z to Z such that f1 + f2 is not surjective.
Show that f : R→ R, given by f(x) = x — [x], is neither one-one nor onto.
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) = 8x3 and g(x) = x1/3.
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 + 1, g (x) = 2x + 3 .
` if f : (-π/2 , π/2)` → R and g : [−1, 1]→ R be defined as f(x) = tan x and g(x) = `sqrt(1 - x^2)` respectively, describe fog and gof.
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).
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.
Write the total number of one-one functions from set A = {1, 2, 3, 4} to set B = {a, b, c}.
If f : R → R defined by f(x) = 3x − 4 is invertible, then write f−1 (x).
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 inverse of the function
\[f : R \to \left\{ x \in R : x < 1 \right\}\] given by
\[f\left( x \right) = \frac{e^x - e^{- x}}{e^x + e^{- x}}\] is
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
Mark the correct alternative in the following question:
Let f : R→ R be defined as, f(x) = \[\begin{cases}2x, if x > 3 \\ x^2 , if 1 < x \leq 3 \\ 3x, if x \leq 1\end{cases}\]
Then, find f( \[-\]1) + f(2) + f(4)
Show that the function f: R → R defined by f(x) = `x/(x^2 + 1)`, ∀ ∈ + R , is neither one-one nor onto
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
Let f: R → R be defined by f(x) = x2 + 1. Then, pre-images of 17 and – 3, respectively, are ______.
Let A be a finite set. Then, each injective function from A into itself is not surjective.
Let f: R → R be the function defined by f(x) = 2x – 3 ∀ x ∈ R. write f–1
Let A = {1, 2, 3, ...n} and B = {a, b}. Then the number of surjections from A into B is ______.
The function f : R → R defined by f(x) = 3 – 4x is ____________.
Let X = {-1, 0, 1}, Y = {0, 2} and a function f : X → Y defiend by y = 2x4, is ____________.
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. Based on the given information, f is best defined as:
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.
- Let R: B → G be defined by R = { (b1,g1), (b2,g2),(b3,g1)}, then 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: N → N be defined by f(x) = x2 is ____________.
Consider a function f: `[0, pi/2] ->` R, given by f(x) = sinx and `g[0, pi/2] ->` R given by g(x) = cosx then f and g are
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.
Difference between the greatest and least value of f(x) = `(1 + (cos^-1x)/π)^2 - (1 + (sin^-1x)/π)^2` 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 ______.
Which real-life situation resembles a one-one idea when no roll number is repeated?
