Advertisements
Advertisements
Question
Let f : R `{- 4/3} `- 43 →">→ R be a function defined as f(x) = `(4x)/(3x +4)` . Show that f : R - `{-4/3}`→ Rang (f) is one-one and onto. Hence, find f -1.
Advertisements
Solution
The function f : R - `{-4/3}→ R - {4/3}` is given by f (x) = `(4x)/(3x +4).` Injectivity: Let `x,y in R - {-4/3} `be such that f (x) = f (y)
⇒ `(4x)/(3x + 4)= (4x)/(3y+4)`
⇒ 4x (3y +4) = 4y (3x +4)
⇒ 12xy + 16x = 12xy +16y
⇒ 16x = 16y
⇒ x = y
Hence, f is one-one function.
Surjectivity: Let y be an arbitrary element of `R - {4/3}.` Then,
f(x) = y
⇒ `(4x)/(3x+4) = y`
⇒ 4x = 3xy +4y
⇒ 4x - 3xy = 4y
⇒ `x = (4y)/(4-3y)`
As `y ∈ R -{4/3} , (4y)/(4-3y) in R.`
Also , `(4y)/(4 - 3y) ≠ -4/3` because `(4y)/(4-3y) = - 4/3 ⇒ 12y = -16 +12y ⇒ 0 = -16,`which is not posssible.
Thus,
`x = (4y)/(4-3y) in R - {- 4/3}` sich that
`f (x) = f ((4x)/(3x +4))= (4((4y)/(4 -3y)))/(3((4y)/(4 -3y))+4) = (16y)/(12y+ 16 - 12y)= (16y)/16` = y , so every element in `R - {4/3} ` has pre-image in `R- {-4/3}.`
Hence, f is onto.
Now,
`x = (4y)/(4 -3y)`
Replacing x by f-1 and y by x, we have
`f^-1 (x) = (4x)/(4 - 3x)`
APPEARS IN
RELATED QUESTIONS
Check the injectivity and surjectivity of the following function:
f : R → R given by f(x) = x2
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.
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):}`
Show that the function f: ℝ → ℝ defined by f(x) = `x/(x^2 + 1), ∀x in R`is neither one-one nor onto. Also, if g: ℝ → ℝ is defined as g(x) = 2x - 1. Find fog(x)
Which of the following functions from A to B are one-one and onto ?
f3 = {(a, x), (b, x), (c, z), (d, z)} ; A = {a, b, c, d,}, B = {x, y, z}.
Let A = [-1, 1]. Then, discuss whether the following functions from A to itself is one-one, onto or bijective : h(x) = x2
Find gof and fog when f : R → R and g : R → R is defined by f(x) = 2x + 3 and g(x) = x2 + 5 .
Find gof and fog when f : R → R and g : R → R is defined by f(x) = x and g(x) = |x| .
Let f : R → R and g : R → R be defined by f(x) = x + 1 and g (x) = x − 1. Show that fog = gof = IR.
Give examples of two functions f : N → N and g : N → N, such that gof is onto but f is not onto.
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 :
g : {5, 6, 7, 8} → {1, 2, 3, 4} with g = {(5, 4), (6, 3), (7, 4), (8, 2)}
If f : R → (0, 2) defined by `f (x) =(e^x - e^(x))/(e^x +e^(-x))+1`is invertible , find f-1.
Let f : [−1, ∞) → [−1, ∞) be given by f(x) = (x + 1)2 − 1, x ≥ −1. Show that f is invertible. Also, find the set S = {x : f(x) = f−1 (x)}.
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.
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 is defined by f(x) = 10 x − 7, then write f−1 (x).
Write the domain of the real function
`f (x) = sqrt([x] - x) .`
Let f : R → R be the function defined by f(x) = 4x − 3 for all x ∈ R Then write f . [NCERT EXEMPLAR]
\[f : R \to R \text{given by} f\left( x \right) = x + \sqrt{x^2} \text{ is }\]
Let the function
\[f : R - \left\{ - b \right\} \to R - \left\{ 1 \right\}\]
\[f\left( x \right) = \frac{x + a}{x + b}, a \neq b .\text{Then},\]
The function \[f : [0, \infty ) \to \text {R given by } f\left( x \right) = \frac{x}{x + 1} is\]
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 =
Let
\[f : R - \left\{ n \right\} \to R\]
The function \[f : R \to R\] defined by
\[f\left( x \right) = 6^x + 6^{|x|}\] is
Let \[f\left( x \right) = x^2 and g\left( x \right) = 2^x\] Then, the solution set of the equation
If \[F : [1, \infty ) \to [2, \infty )\] is given by
\[f\left( x \right) = x + \frac{1}{x}, then f^{- 1} \left( x \right)\]
Let
\[f : R \to R\] be given by \[f\left( x \right) = x^2 - 3\] Then, \[f^{- 1}\] is given by
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)
Mark the correct alternative in the following question:
Let A = {1, 2, ... , n} and B = {a, b}. Then the number of subjections from A into B is
If A = {a, b, c, d} and f = {a, b), (b, d), (c, a), (d, c)}, show that f is one-one from A onto A. Find f–1
Show that the function f: R → R defined by f(x) = `x/(x^2 + 1)`, ∀ ∈ + R , is neither one-one nor onto
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.
For sets A, B and C, let f: A → B, g: B → C be functions such that g o f is injective. Then both f and g are injective functions.
The function f : A → B defined by f(x) = 4x + 7, x ∈ R is ____________.
The number of bijective functions from set A to itself when A contains 106 elements is ____________.
Let n(A) = 4 and n(B) = 6, Then the number of one – one functions from 'A' to 'B' is:
The solution set of the inequation log1/3(x2 + x + 1) + 1 > 0 is ______.
