Advertisements
Advertisements
प्रश्न
Let
\[f : R \to R\] be a function defined by
विकल्प
f is a bijection
f is an injection only
f is surjection on only
f is neither an injection nor a surjection
Advertisements
उत्तर
f is neither an injection nor a surjection
\[f : R \to R\]
\[f\left( x \right) = \frac{e^{|x|} - e^{- x}}{e^x + e^{- x}}\]
\[\text{For } x = - 2 \text{ and} - 3 \in R \]
\[f( - 2) = \frac{e^\left| - 2 \right| - e^2}{e^{- 2} + e^2}\]
\[ = \frac{e^2 - e^2}{e^{- 2} + e^2}\]
\[ = 0\]
\[\text{& } f( - 3) = \frac{e^\left| - 3 \right| - e^3}{e^{- 3} + e^3}\]
\[ = \frac{e^3 - e^3}{e^{- 3} + e^3}\]
\[ = 0\] \[\text{Hence, for different values of x we are getting same values of f }(x)\]
\[\text{That means , the given function is many one} . \]
Therefore, this function is not injective.
\[ \text{For } x < 0\]
\[f (x ) = 0\]
\[\text{ For } x > 0\]
\[f(x) = \frac{e^x - e^{- x}}{e^x + e^{- x}}\]
\[ = \frac{e^x + e^{- x}}{e^x + e^{- x}} - \frac{2 e^{- x}}{e^x + e^{- x}}\]
\[ = 1 - \frac{2 e^{- x}}{e^x + e^{- x}}\]
\[\text{The value of } \frac{2 e^{- x}}{e^x + e^{- x}} \text{is always positive} . \]
\[\text{Therefore, the value of} f(x) \text{is always less than} 1\]
\[\text{Numbers more than 1 are not included in the range but they are included in codomain} . \]
\[\text{As the codomain is } R . \]
\[ \therefore \text{Codomain} \neq \text{Range}\]
\[\text{Hence, the given function is not onto} . \]
Therefore, this function is not surjective .
APPEARS IN
संबंधित प्रश्न
Check the injectivity and surjectivity of the following function:
f : Z → Z given by f(x) = x2
Let f: R → R be defined as f(x) = 10x + 7. Find the function g: R → R such that g o f = f o g = 1R.
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):}`
Which of the following functions from A to B are one-one and onto?
f2 = {(2, a), (3, b), (4, c)} ; A = {2, 3, 4}, B = {a, b, c}
Prove that the function f : N → N, defined by f(x) = x2 + x + 1, is one-one but not onto
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
Give examples of two one-one functions f1 and f2 from R to R, such that f1 + f2 : R → R. defined by (f1 + f2) (x) = f1 (x) + f2 (x) is not one-one.
Show that if f1 and f2 are one-one maps from R to R, then the product f1 × f2 : R → R defined by (f1 × f2) (x) = f1 (x) f2 (x) need not be one - one.
Find fog (2) and gof (1) when : f : R → R ; f(x) = x2 + 8 and g : R → R; g(x) = 3x3 + 1.
Find fog and gof if : f (x) = ex g(x) = loge x .
Find fog and gof if : f (x) = |x|, g (x) = sin x .
Find fog and gof if : f(x)= x + 1, g (x) = 2x + 3 .
Let f be a real function given by f (x)=`sqrt (x-2)`
Find each of the following:
(i) fof
(ii) fofof
(iii) (fofof) (38)
(iv) f2
Also, show that fof ≠ `f^2` .
If f, g : R → R be two functions defined as f(x) = |x| + x and g(x) = |x|- x, ∀x∈R" .Then find fog and gof. Hence find fog(–3), fog(5) and gof (–2).
State with reason whether the following functions have inverse :
f : {1, 2, 3, 4} → {10} with f = {(1, 10), (2, 10), (3, 10), (4, 10)}
If f : A → A, g : A → A are two bijections, then prove that fog is an injection ?
If f : R → R is defined by f(x) = x2, write f−1 (25)
If f(x) = x + 7 and g(x) = x − 7, x ∈ R, write fog (7).
If f : R → R be defined by f(x) = (3 − x3)1/3, then find fof (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]
If the mapping f : {1, 3, 4} → {1, 2, 5} and g : {1, 2, 5} → {1, 3}, given by f = {(1, 2), (3, 5), (4, 1)} and g = {(2, 3), (5, 1), (1, 3)}, then write fog. [NCERT EXEMPLAR]
The function f : R → R defined by
`f (x) = 2^x + 2^(|x|)` is
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 function \[f : [0, \infty ) \to \text {R given by } f\left( x \right) = \frac{x}{x + 1} is\]
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 \[f : R \to R\] defined by
\[f\left( x \right) = 6^x + 6^{|x|}\] is
If \[f\left( x \right) = \sin^2 x\] and the composite function \[g\left( f\left( x \right) \right) = \left| \sin x \right|\] then g(x) is equal to
Let
\[f : R \to R\] be given by \[f\left( x \right) = x^2 - 3\] Then, \[f^{- 1}\] is given by
If f(x) = `(x+3)/(4x−5) , "g"(x) = (3+5x)/(4x−1)` then verify that `("fog") (x)` = x.
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
For sets A, B and C, let f: A → B, g: B → C be functions such that g o f is surjective. Then g is surjective.
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)}
Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:
k(x) = x2
The function f : R → R defined by f(x) = 3 – 4x is ____________.
The mapping f : N → N is given by f(n) = 1 + n2, n ∈ N when N is the set of natural numbers 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 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: R → R be defined by f(x) = x2 is:
If f: [0, 1]→[0, 1] is defined by f(x) = `(x + 1)/4` and `d/(dx) underbrace(((fofof......of)(x)))_("n" "times")""|_(x = 1/2) = 1/"m"^"n"`, m ∈ N, then the value of 'm' is ______.
The function f(x) = [x], where [x] denotes the greatest integer less than or equal to x; is continuous at ______.
