Advertisements
Advertisements
प्रश्न
Let f : R → R be the function defined by f(x) = 4x − 3 for all x ∈ R Then write f . [NCERT EXEMPLAR]
Advertisements
उत्तर
We have,
f : R → R is the function defined by f(x) = 4x −3 for all x ∈ R
\[y = 4x - 3\]
\[ \Rightarrow 4x = y + 3\]
\[ \Rightarrow x = \frac{y + 3}{4}\]
\[So, f^{- 1} \left( y \right) = \frac{y + 3}{4}\]
\[or, f^{- 1} \left( x \right) = \frac{x + 3}{4}\]
APPEARS IN
संबंधित प्रश्न
Show that the signum function f : R → R, given by
`f(x) = {(1", if" x > 0), (0", if" x = 0), (-1", if" x < 0):}`
is neither one-one nor onto.
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.
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}
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 : Q − {3} → Q, defined by `f (x) = (2x +3)/(x-3)`
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = `x/(x^2 +1)`
Given A = {2, 3, 4}, B = {2, 5, 6, 7}. Construct an example of each of the following:
(i) an injective map from A to B
(ii) a mapping from A to B which is not injective
(iii) a mapping from A to B.
If f : A → B and g : B → C are onto functions, show that gof is a onto function.
If f(x) = sin x and g(x) = 2x be two real functions, then describe gof and fog. Are these equal functions?
Let f be any real function and let g be a function given by g(x) = 2x. Prove that gof = f + f.
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` .
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)}
State with reason whether the following functions have inverse :
g : {5, 6, 7, 8} → {1, 2, 3, 4} with g = {(5, 4), (6, 3), (7, 4), (8, 2)}
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).
Let A = R - {3} and B = R - {1}. Consider the function f : A → B defined by f(x) = `(x-2)/(x-3).`Show that f is one-one and onto and hence find f-1.
[CBSE 2012, 2014]
Let A and B be two sets, each with a finite number of elements. Assume that there is an injective map from A to B and that there is an injective map from B to A. Prove that there is a bijection from A to B.
If f : A → A, g : A → A are two bijections, then prove that fog is a surjection ?
Which of the following graphs represents a one-one function?
Write the domain of the real function
`f (x) = sqrt([x] - x) .`
Write whether f : R → R, given by `f(x) = x + sqrtx^2` is one-one, many-one, onto or into.
If a function g = {(1, 1), (2, 3), (3, 5), (4, 7)} is described by g(x) = \[\alpha x + \beta\] then find the values of \[\alpha\] and \[ \beta\] . [NCERT EXEMPLAR]
If f(x) = 4 −( x - 7)3 then write f-1 (x).
\[f : R \to R \text{given by} f\left( x \right) = x + \sqrt{x^2} \text{ is }\]
\[f : A \to \text{B given by } 3^{ f\left( x \right)} + 2^{- x} = 4\] is a bijection, then
Let
\[f : [2, \infty ) \to X\] be defined by
\[f\left( x \right) = 4x - x^2\] Then, f is invertible if X =
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 \[f\left(x\right) = x^3\] be a function with domain {0, 1, 2, 3}. Then domain of \[f^{-1}\] is ______.
Let N be the set of natural numbers and the function f: N → N be defined by f(n) = 2n + 3 ∀ n ∈ N. Then f 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
k = {(1,4), (2, 5)}
Let f: R → R be the functions defined by f(x) = x3 + 5. Then f–1(x) is ______.
Let f : R → R be defind by f(x) = `1/"x" AA "x" in "R".` Then f is ____________.
Range of `"f"("x") = sqrt((1 - "cos x") sqrt ((1 - "cos x")sqrt ((1 - "cos x")....infty))`
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:
The domain of the function `cos^-1((2sin^-1(1/(4x^2-1)))/π)` is ______.
Number of integral values of x satisfying the inequality `(3/4)^(6x + 10 - x^2) < 27/64` is ______.
`x^(log_5x) > 5` implies ______.
For x ∈ R, x ≠ 0, let f0(x) = `1/(1 - x)` and fn+1 (x) = f0(fn(x)), n = 0, 1, 2, .... Then the value of `f_100(3) + f_1(2/3) + f_2(3/2)` is equal to ______.
