Advertisements
Advertisements
प्रश्न
Consider the function f : R+ → [-9 , ∞ ]given by f(x) = 5x2 + 6x - 9. Prove that f is invertible with f -1 (y) = `(sqrt(54 + 5y) -3)/5` [CBSE 2015]
Advertisements
उत्तर
We have ,
f (x) = 5x2+ 6x − 9
Let y = 5x2+ 6x − 9
` = 5 (x^2 + 6/5x - 9/5)`
` = 5(x^2 + 2 xx x xx 3/5 + 9 /25 - 9/25 - 9/5)`
`= (( x + 3/5)^2 - 9/25 - 9/25)`
`=(x+ 3/5)^2 - 9/5 - 9 `
`= 5 (x + 3/5)^2 - 54/5`
⇒ `y + 54/5 = 5 (x+3/5)^2`
⇒ `(5y + 54)/25 (x + 3/5)^2`
⇒ `sqrt (5y +54)/25 = x +3/5`
⇒ `x = sqrt (5y +54)/5 - 3/5`
⇒ `x = (sqrt (5y +54)-3)/5 `
Let g (y) =` (sqrt(5y +54) -3)/5`
Now,
fog (y) = f (g (y))
= f `((sqrt (5y+54)-3)/5)`
= 5 `((sqrt (5y+54)-3)/5)^2 + 6 ((sqrt (5y+54)-3)/5) = - 9 `
`= 5 ((5y + 54 +9 - 6 sqrt (5y +54))/25) + ((6 sqrt(5y + 54) -18)/5) -9`
`= (5y + 63 - 6 sqrt (5y + 54))/5 +(6 sqrt (5y + 54)- 18)/5 -9`
=` (5y + 63 - 18 - 45) /5`
= y
= IY, Identity function
Also, gof (x) = g (f(x))
= g (5x2 + 6x - 9 )
`= (sqrt(5(5x^2 + 6x - 9)+ 54)-3)/5`
`= (sqrt(25x^2 + 30x - 45 +54) -3)/5`
`=(sqrt(25 x^2 + 30x + 9) -3)/5`
`= (sqrt((5x + 3)^2) - 3)/5`
`= (5x +3 -3)/5`
= x
= IX , Identity function
So, f is invertible .
Also, `f^-1 (y) = g (y) = (sqrt(5y +54) -3)/5`
APPEARS IN
संबंधित प्रश्न
Classify the following function as injection, surjection or bijection :
f : Z → Z, defined by f(x) = x − 5
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = sin2x + cos2x
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 function from A to itself is one-one, onto or bijective : `f (x) = x/2`
Let A = [-1, 1]. Then, discuss whether the following function from A to itself is one-one, onto or bijective : g(x) = |x|
Let A = [-1, 1]. Then, discuss whether the following functions from A to itself is one-one, onto or bijective : h(x) = x2
Give examples of two surjective functions f1 and f2 from Z to Z such that f1 + f2 is not surjective.
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.
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.
Consider f : R → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f.
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 = {x &epsis; R | −1 ≤ x ≤ 1} and let f : A → A, g : A → A be two functions defined by f(x) = x2 and g(x) = sin (π x/2). Show that g−1 exists but f−1 does not exist. Also, find g−1.
Write whether f : R → R, given by `f(x) = x + sqrtx^2` is one-one, many-one, onto or into.
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]
\[f : R \to R \text{given by} f\left( x \right) = x + \sqrt{x^2} \text{ is }\]
If a function\[f : [2, \infty )\text{ to B defined by f}\left( x \right) = x^2 - 4x + 5\] is a bijection, then B =
The function
\[f : R \to R\] defined by\[f\left( x \right) = \left( x - 1 \right) \left( x - 2 \right) \left( x - 3 \right)\]
(a) one-one but not onto
(b) onto but not one-one
(c) both one and onto
(d) neither one-one nor onto
The function f : [-1/2, 1/2, 1/2] → [-π /2,π/2], defined by f (x) = `sin^-1` (3x - `4x^3`), is
If \[F : [1, \infty ) \to [2, \infty )\] is given by
\[f\left( x \right) = x + \frac{1}{x}, then f^{- 1} \left( x \right)\]
If \[f : R \to \left( - 1, 1 \right)\] is defined by
\[f\left( x \right) = \frac{- x|x|}{1 + x^2}, \text{ then } f^{- 1} \left( x \right)\] equals
Mark the correct alternative in the following question:
Let f : R → R be given by f(x) = tanx. Then, f-1(1) is
Mark the correct alternative in the following question:
Let f : R \[-\] \[\left\{ \frac{3}{5} \right\}\] \[\to\] R be defined by f(x) = \[\frac{3x + 2}{5x - 3}\] Then,
Write about strlen() function.
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 f: R → R be defined by f(x) = 3x – 4. Then f–1(x) is given by ______.
Are the following set of ordered pairs functions? If so, examine whether the mapping is injective or surjective.
{(x, y): x is a person, y is the mother of x}
Let f: R → R be given by f(x) = tan x. Then f–1(1) is ______.
The function f : R → R given by f(x) = x3 – 1 is ____________.
Let f : R `->` R be a function defined by f(x) = x3 + 4, then f is ______.
Sherlin and Danju are playing Ludo at home during Covid-19. While rolling the dice, Sherlin’s sister Raji observed and noted the possible outcomes of the throw every time belongs to set {1,2,3,4,5,6}. Let A be the set of players while B be the set of all possible outcomes.
A = {S, D}, B = {1,2,3,4,5,6}
- Raji wants to know the number of functions from A to B. How many number of functions are possible?
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: {1,2,3,....} → {1,4,9,....} be defined by f(x) = x2 is ____________.
A function f: x → y is/are called onto (or surjective) if x under f.
Let f: R→R be a continuous function such that f(x) + f(x + 1) = 2, for all x ∈ R. If I1 = `int_0^8f(x)dx` and I2 = `int_(-1)^3f(x)dx`, then the value of I1 + 2I2 is equal to ______.
The solution set of the inequation log1/3(x2 + x + 1) + 1 > 0 is ______.
Difference between the greatest and least value of f(x) = `(1 + (cos^-1x)/π)^2 - (1 + (sin^-1x)/π)^2` is ______.
Let A = {1, 2, 3, ..., 10} and f : A `rightarrow` A be defined as
f(k) = `{{:(k + 1, if k "is odd"),( k, if k "is even"):}`.
Then the number of possible functions g : A `rightarrow` A such that gof = f is ______.
