Advertisements
Advertisements
प्रश्न
Consider f : R+ → [−5, ∞) given by f(x) = 9x2 + 6x − 5. Show that f is invertible with `f^-1 (x) = (sqrt (x +6)-1)/3 .`
Advertisements
उत्तर
Injectivity of f :
Let x and y be two elements of domain` (R^+)`, such that
f(x)=f(y)
⇒ 9x2+6x−5=9y2+ 6y − 5
⇒ 9x2+6x=9y2+6y
⇒ x = y (As, x, y ∈ `R^+`)
So, f is one-one.
Surjectivity of f:
Let y is in the co domain (Q) such that f(x) = y
⇒ 9x2 + 6x - 5 = y
⇒ 9x2 +6x = y + 5
⇒ 9x2 + 6x +1 = y +6 (Adding 1 on both sides )
⇒ (3x +1)2 = y + 6
⇒ `3x +1 = sqrt(y + 6)`
⇒ `3x = sqrt (y + 6) -1`
⇒ `x = (sqrt (y + 6)-1)/3 in R^+` (domain)
f is onto.
So, f is a bijection and hence, it is invertible.
Finding `f^-1`
Let f−1(x) = y ...(1)
⇒ x = f (y)
⇒ x = 9y2+ 6y − 5
⇒ x + 5 = 9y2+6y
⇒ x + 6= 9y2+ 6y + 1 (adding 1 on both sides)
⇒ x + 6 = ( 3y + 1 )2
⇒3y+1=`sqrt(x +6)`
⇒ `3y = sqrt (x +6) -1`
⇒ `y = (sqrt (x+6)-1)/3`
`So, f^-1 (x) (sqrt (x-6)-1)/3 ` [from (1)]
APPEARS IN
संबंधित प्रश्न
In the following case, state whether the function is one-one, onto or bijective. Justify your answer.
f : R → R defined by f(x) = 1 + x2
Give an example of a function which is not one-one but onto ?
Give an example of a function which is neither one-one nor onto ?
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}
Classify the following function as injection, surjection or bijection : f : Z → Z given by f(x) = x2
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 : N → N be defined by
`f(n) = { (n+ 1, if n is odd),( n-1 , if n is even):}`
Show that f is a bijection.
[CBSE 2012, NCERT]
Let f = {(1, −1), (4, −2), (9, −3), (16, 4)} and g = {(−1, −2), (−2, −4), (−3, −6), (4, 8)}. Show that gof is defined while fog is not defined. Also, find gof.
Give examples of two functions f : N → N and g : N → N, such that gof is onto but f is not onto.
If f(x) = 2x + 5 and g(x) = x2 + 1 be two real functions, then describe each of the following functions:
(1) fog
(2) gof
(3) fof
(4) f2
Also, show that fof ≠ f2
` if f : (-π/2 , π/2)` → R and g : [−1, 1]→ R be defined as f(x) = tan x and g(x) = `sqrt(1 - x^2)` respectively, describe fog and gof.
Consider f : {1, 2, 3} → {a, b, c} and g : {a, b, c} → {apple, ball, cat} defined as f (1) = a, f (2) = b, f (3) = c, g (a) = apple, g (b) = ball and g (c) = cat. Show that f, g and gof are invertible. Find f−1, g−1 and gof−1and show that (gof)−1 = f −1o g−1
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.
If f : A → A, g : A → A are two bijections, then prove that fog is an injection ?
If f : R → R is defined by f(x) = x2, find f−1 (−25).
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) = 1/(sqrt([x] - x)`.
What is the range of the function
`f (x) = ([x - 1])/(x -1) ?`
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]
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
Let A = R − (2) and B = R − (1). If f: A ⟶ B is a function defined by`"f(x)"=("x"-1)/("x"-2),` how that f is one-one and onto. Hence, find f−1.
Consider the set A containing n elements. Then, the total number of injective functions from A onto itself is ______
Let A = R – {3}, B = R – {1}. Let f: A → B be defined by f(x) = `(x - 2)/(x - 3)` ∀ x ∈ A . Then show that f is bijective.
Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:
g(x) = |x|
If the set A contains 5 elements and the set B contains 6 elements, then the number of one-one and onto mappings from A to B is ______.
Which of the following functions from Z into Z are bijections?
Let f: R → R be given by f(x) = tan x. Then f–1(1) is ______.
Let A = {0, 1} and N be the set of natural numbers. Then the mapping f: N → A defined by f(2n – 1) = 0, f(2n) = 1, ∀ n ∈ N, is onto.
Let f : R → R be defind by f(x) = `1/"x" AA "x" in "R".` Then f is ____________.
Let g(x) = x2 – 4x – 5, then ____________.
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. Based on the given information, f is best defined as:
An organization conducted a bike race under 2 different categories-boys and girls. Totally there were 250 participants. Among all of them finally, three from Category 1 and two from Category 2 were selected for the final race. Ravi forms two sets B and G with these participants for his college project. Let B = {b1,b2,b3} G={g1,g2} where B represents the set of boys selected and G the set of girls who were selected for the final race.
Ravi decides to explore these sets for various types of relations and functions.
- Let R: B → G be defined by R = { (b1,g1), (b2,g2),(b3,g1)}, then R 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 solution set of the inequation log1/3(x2 + x + 1) + 1 > 0 is ______.
Let x is a real number such that are functions involved are well defined then the value of `lim_(t→0)[max{(sin^-1 x/3 + cos^-1 x/3)^2, min(x^2 + 4x + 7)}]((sin^-1t)/t)` where [.] is greatest integer function and all other brackets are usual brackets.
The domain of function is f(x) = `sqrt(-log_0.3(x - 1))/sqrt(x^2 + 2x + 8)` is ______.
Let a function `f: N rightarrow N` be defined by
f(n) = `{:[(2n",", n = 2"," 4"," 6"," 8","......),(n - 1",", n = 3"," 7"," 11"," 15","......),((n + 1)/2",", n = 1"," 5"," 9"," 13","......):}`
then f is ______.
Find the domain of sin–1 (x2 – 4).
