Advertisements
Advertisements
प्रश्न
Let f : R → R be defined as `f (x) = (2x - 3)/4.` write fo f-1 (1) .
Advertisements
उत्तर
\[Let f^{- 1} \left( x \right) = y . . . \left( 1 \right)\]
\[ \Rightarrow f\left( y \right) = x\]
\[ \Rightarrow \frac{2y - 3}{4} = x\]
\[ \Rightarrow 2y - 3 = 4x\]
\[ \Rightarrow 2y = 4x + 3\]
\[ \Rightarrow y = \frac{4x + 3}{2}\]
\[ \Rightarrow f^{- 1} \left( x \right) = \frac{4x + 3}{2} [\text{ from}\left( 1 \right)]\]
\[ \Rightarrow f^{- 1} \left( x \right) = \frac{4x + 3}{2}\]
\[ \therefore \left( fo f^{- 1} \right)\left( 1 \right) = f\left( \frac{4\left( 1 \right) + 3}{2} \right) = f\left( \frac{7}{2} \right) = \frac{2\left( \frac{7}{2} \right) - 3}{4} = \frac{7 - 3}{4} = \frac{4}{4} = 1\]
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 signum function f : R → R, given by
`f(x) = {(1", if" x > 0), (0", if" x = 0), (-1", if" x < 0):}`
is neither one-one nor onto.
In the following case, state whether the function is one-one, onto or bijective. Justify your answer.
f : R → R defined by f(x) = 3 – 4x
Let f : R → R be defined as f(x) = x4. Choose the correct answer.
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
Let A = [-1, 1]. Then, discuss whether the following functions from A to itself is one-one, onto or bijective : h(x) = x2
Set of ordered pair of a function? If so, examine whether the mapping is injective or surjective :{(x, y) : x is a person, y is the mother of x}
Set of ordered pair of a function ? If so, examine whether the mapping is injective or surjective :{(a, b) : a is a person, b is an ancestor of a}
Give examples of two one-one functions f1 and f2 from R to R, such that f1 + f2 : R → R. defined by (f1 + f2) (x) = f1 (x) + f2 (x) is not one-one.
If f : A → B and g : B → C are one-one functions, show that gof is a one-one function.
Find fog and gof if : f (x) = x+1, g(x) = `e^x`
.
State with reason whether the following functions have inverse :
f : {1, 2, 3, 4} → {10} with f = {(1, 10), (2, 10), (3, 10), (4, 10)}
Consider f : R → R+ → [4, ∞) given by f(x) = x2 + 4. Show that f is invertible with inverse f−1 of f given by f−1 `(x)= sqrt (x-4)` where R+ is the set of all non-negative real numbers.
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) = 10 x − 7, then write f−1 (x).
Write whether f : R → R, given by `f(x) = x + sqrtx^2` is one-one, many-one, onto or into.
If f : R → R be defined by f(x) = (3 − x3)1/3, then find fof (x).
If a function g = {(1, 1), (2, 3), (3, 5), (4, 7)} is described by g(x) = \[\alpha x + \beta\] then find the values of \[\alpha\] and \[ \beta\] . [NCERT EXEMPLAR]
If a function\[f : [2, \infty )\text{ to B defined by f}\left( x \right) = x^2 - 4x + 5\] is a bijection, then B =
Which of the following functions from
\[A = \left\{ x \in R : - 1 \leq x \leq 1 \right\}\]
Let \[f\left( x \right) = x^2 and g\left( x \right) = 2^x\] Then, the solution set of the equation
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,
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
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 ______.
For sets A, B and C, let f: A → B, g: B → C be functions such that g o f is surjective. Then g is surjective.
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 X = {1, 2, 3}and Y = {4, 5}. Find whether the following subset of X ×Y are function from X to Y or not
k = {(1,4), (2, 5)}
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.
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 ______.
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?
'If 'f' is a linear function satisfying f[x + f(x)] = x + f(x), then f(5) can be equal to:
Let n(A) = 4 and n(B) = 6, Then the number of one – one functions from 'A' to 'B' is:
Consider a function f: `[0, pi/2] ->` R, given by f(x) = sinx and `g[0, pi/2] ->` R given by g(x) = cosx then f and g are
If f: R→R is a function defined by f(x) = `[x - 1]cos((2x - 1)/2)π`, where [ ] denotes the greatest integer function, then f is ______.
If A = {x ∈ R: |x – 2| > 1}, B = `{x ∈ R : sqrt(x^2 - 3) > 1}`, C = {x ∈ R : |x – 4| ≥ 2} and Z is the set of all integers, then the number of subsets of the set (A ∩ B ∩ C) C ∩ Z is ______.
If f : R `rightarrow` R is defined by `f(x) = (2x - 7)/4`, show that f(x) is one-one and onto.
