Advertisements
Advertisements
प्रश्न
Show that the function f in `A=R-{2/3} ` defined as `f(x)=(4x+3)/(6x-4)` is one-one and onto hence find f-1
Advertisements
उत्तर
The given function is
`f(x)=(4x+3)/(6x-4)`
`Let f(x_1)=f(x_2)`
`(4x_1+3)/(6x_1-4)=(4x_2+3)/(6x_2-4)`
⇒ 24x1x2 − 16x1 + 18x2 − 12 = 24x1x2 + 18x1 − 16x2 − 12
⇒18x2 + 16x2 = 18x1 + 16x1
⇒34x2 = 34x1⇒x1= x2
Therefore f(x) is one − one.
Since, `(4x+3)/(6x-4) ` is a real number, therefore, for every y in the co–domain (f), there exists a number x in `R-{2/3}` such that
`f(x)=y=(4x+3)/(6x-4)`
Therefore, f(x) is onto
Now let `y=(4x+3)/(6x-4)`
`6xy-4y=4x+3`
`x=(4y+3)/(6y-4)`
`f^(-1)(x)=(4x+3)/(6x-4)`
APPEARS IN
संबंधित प्रश्न
Check the injectivity and surjectivity of the following function:
f : Z → Z given by f(x) = x3
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.
Show that the function f : R → R given by f(x) = x3 is injective.
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = sinx
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = sin2x + cos2x
Show that the logarithmic function f : R0+ → R given by f (x) loga x ,a> 0 is a bijection.
If A = {1, 2, 3}, show that a onto function f : A → A must be one-one.
Find the number of all onto functions from the set A = {1, 2, 3, ..., n} to itself.
Let f be any real function and let g be a function given by g(x) = 2x. Prove that gof = f + f.
If A = {1, 2, 3, 4} and B = {a, b, c, d}, define any four bijections from A to B. Also give their inverse functions.
If A = {a, b, c} and B = {−2, −1, 0, 1, 2}, write the total number of one-one functions from A to B.
If f : R → R be defined by f(x) = (3 − x3)1/3, then find fof (x).
\[f : A \to \text{B given by } 3^{ f\left( x \right)} + 2^{- x} = 4\] is a bijection, then
The function
The function \[f : R \to R\] defined by
\[f\left( x \right) = 6^x + 6^{|x|}\] is
If \[g \left( f \left( x \right) \right) = \left| \sin x \right| \text{and} f \left( g \left( x \right) \right) = \left( \sin \sqrt{x} \right)^2 , \text{then}\]
The distinct linear functions that map [−1, 1] onto [0, 2] are
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
Mark the correct alternative in the following question:
If the set A contains 7 elements and the set B contains 10 elements, then the number one-one functions from A to B is
Let R be the set of real numbers and f: R → R be the function defined by f(x) = 4x + 5. Show that f is invertible and find f–1.
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 ______.
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 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 the function f: R → R be defined by f(x) = cosx, ∀ x ∈ R. Show that f is neither one-one nor onto
The mapping f : N → N is given by f(n) = 1 + n2, n ∈ N when N is the set of natural numbers 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.
- The function f: Z → Z defined by f(x) = x2 is ____________.
The trigonometric equation tan–1x = 3tan–1 a has solution for ______.
