Advertisements
Advertisements
Question
Let
\[f : R \to R\] be a function defined by
Options
f is a bijection
f is an injection only
f is surjection on only
f is neither an injection nor a surjection
Advertisements
Solution
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
RELATED QUESTIONS
Let f : R → R be defined as f(x) = 3x. Choose the correct answer.
Show that the function f : R → {x ∈ R : –1 < x < 1} defined by f(x) = `x/(1 + |x|)`, x ∈ R is one-one and onto function.
Show that the function f : R → R given by f(x) = x3 is injective.
Give an example of a function which is not one-one but onto ?
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = 3 − 4x
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = `x/(x^2 +1)`
Let A = [-1, 1]. Then, discuss whether the following functions from A to itself is one-one, onto or bijective : h(x) = x2
Show that f : R→ R, given by f(x) = x — [x], is neither one-one nor onto.
Give examples of two functions f : N → N and g : N → N, such that gof is onto but f is not onto.
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+1, g(x) = `e^x`
.
Let A = {1, 2, 3, 4}; B = {3, 5, 7, 9}; C = {7, 23, 47, 79} and f : A → B, g : B → C be defined as f(x) = 2x + 1 and g(x) = x2 − 2. Express (gof)−1 and f−1 og−1 as the sets of ordered pairs and verify that (gof)−1 = f−1 og−1.
Consider f : R → R+ → [4, ∞) given by f(x) = x2 + 4. Show that f is invertible with inverse f−1 of f given by f−1 `(x)= sqrt (x-4)` where R+ is the set of all non-negative real numbers.
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.
If f : R → (−1, 1) defined by `f (x) = (10^x- 10^-x)/(10^x + 10 ^-x)` is invertible, find f−1.
If f : A → A, g : A → A are two bijections, then prove that fog is a surjection ?
If A = {1, 2, 3} and B = {a, b}, write the total number of functions from A to B.
If f : R → R is defined by f(x) = x2, write f−1 (25)
If f : R → R is defined by f(x) = x2, find f−1 (−25).
If f : R → R, g : R → are given by f(x) = (x + 1)2 and g(x) = x2 + 1, then write the value of fog (−3).
If f(x) = x + 7 and g(x) = x − 7, x ∈ R, write fog (7).
What is the range of the function
`f (x) = ([x - 1])/(x -1) ?`
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]
If f(x) = 4 −( x - 7)3 then write f-1 (x).
\[f : A \to \text{B given by } 3^{ f\left( x \right)} + 2^{- x} = 4\] is a bijection, then
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
Let A = R − (2) and B = R − (1). If f: A ⟶ B is a function defined by`"f(x)"=("x"-1)/("x"-2),` how that f is one-one and onto. Hence, find f−1.
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
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 f: R → R be the functions defined by f(x) = x3 + 5. Then f–1(x) is ______.
The smallest integer function f(x) = [x] is ____________.
The function f : R → R given by f(x) = x3 – 1 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 ____________.
If f: R → R given by f(x) =(3 − x3)1/3, find f0f(x)
A function f: x → y is/are called onto (or surjective) if x under f.
Function f: R → R, defined by f(x) = `x/(x^2 + 1)` ∀ x ∈ R is not
The domain of function is f(x) = `sqrt(-log_0.3(x - 1))/sqrt(x^2 + 2x + 8)` is ______.
The function defined by \[\mathrm{f}(x)=\frac{2x+3}{3x+4},x\neq-\frac{4}{3}\] is
