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
Check the injectivity and surjectivity of the following function:
f : Z → Z given by f(x) = x2
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)
If the function `f(x) = sqrt(2x - 3)` is invertible then find its inverse. Hence prove that `(fof^(-1))(x) = x`
Classify the following function as injection, surjection or bijection : f : Z → Z given by f(x) = x3
Let A = [-1, 1]. Then, discuss whether the following function from A to itself is one-one, onto or bijective : g(x) = |x|
Give examples of two functions f : N → Z and g : Z → Z, such that gof is injective but gis not injective.
Find fog and gof if : f (x) = x2 g(x) = cos x .
If f(x) = |x|, prove that fof = f.
Let f, g, h be real functions given by f(x) = sin x, g (x) = 2x and h (x) = cos x. Prove that fog = go (fh).
Consider f : R → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f.
If f : R → (−1, 1) defined by `f (x) = (10^x- 10^-x)/(10^x + 10 ^-x)` is invertible, find f−1.
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).
Let f : R → R+ be defined by f(x) = ax, a > 0 and a ≠ 1. Write f−1 (x).
Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. State whether f is one-one or not.
Let A = {a, b, c, d} and f : A → A be given by f = {( a,b ),( b , d ),( c , a ) , ( d , c )} write `f^-1`. [NCERT EXEMPLAR]
Let M be the set of all 2 × 2 matrices with entries from the set R of real numbers. Then, the function f : M→ R defined by f(A) = |A| for every A ∈ M, is
Which of the following functions from
to itself are bijections?
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 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
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
A function f: R→ R defined by f(x) = `(3x) /5 + 2`, x ∈ R. Show that f is one-one and onto. Hence find f−1.
Which function is used to check whether a character is alphanumeric or not?
Write about strlen() function.
Write about strcmp() 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.
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 C be the set of complex numbers. Prove that the mapping f: C → R given by f(z) = |z|, ∀ z ∈ C, is neither one-one nor onto.
Using the definition, prove that the function f: A→ B is invertible if and only if f is both one-one and onto
Let f : R → R be defind by f(x) = `1/"x" AA "x" in "R".` Then f is ____________.
The domain of the function `"f"("x") = 1/(sqrt ({"sin x"} + {"sin" ( pi + "x")}))` where {.} denotes fractional part, is
The function f: R → R defined as f(x) = x3 is:
Let f: R → R defined by f(x) = 3x. Choose the correct answer
Let a and b are two positive integers such that b ≠ 1. Let g(a, b) = Number of lattice points inside the quadrilateral formed by lines x = 0, y = 0, x = b and y = a. f(a, b) = `[a/b] + [(2a)/b] + ... + [((b - 1)a)/b]`, then the value of `[(g(101, 37))/(f(101, 37))]` is ______.
(Note P(x, y) is lattice point if x, y ∈ I)
(where [.] denotes greatest integer function)
The domain of function is f(x) = `sqrt(-log_0.3(x - 1))/sqrt(x^2 + 2x + 8)` is ______.
If f : R `rightarrow` R is defined by `f(x) = (2x - 7)/4`, show that f(x) is one-one and onto.
