Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
Here the function f : R → R is defined as f (x) = 4x + 5 = y (say). Then
4x = y – 5 or x = `(y - 5)/4`.
This leads to a function g: R → R defined as
g(y) = `(y - 5)/4`.
Therefore, (gof) (x) = g(f(x) = g(4x + 5)
= `(4x + 5 - 5)/4`
= x
or
gof = IR
Similarly (fog) (y) = f(g(y))
= `f((y - 5)/4)`
= `4((y - 5)/4) + 5`
= y
or
fog = IR
Hence f is invertible and f-1 = g which is given by `f^-1 (x) = (x - 5)/4`
APPEARS IN
संबंधित प्रश्न
Show that the function f in `A=R-{2/3} ` defined as `f(x)=(4x+3)/(6x-4)` is one-one and onto hence find f-1
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}.
Classify the following function as injection, surjection or bijection : f : Z → Z given by f(x) = x3
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) = x3 + 1
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}
Give examples of two surjective functions f1 and f2 from Z to Z such that f1 + f2 is not surjective.
Give examples of two functions f : N → Z and g : Z → Z, such that gof is injective but gis not injective.
If f : A → B and g : B → C are one-one functions, show that gof is a one-one function.
If f : A → B and g : B → C are onto functions, show that gof is a onto function.
If f(x) = |x|, prove that fof = f.
Consider f : R → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f.
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 : R → (0, 2) defined by `f (x) =(e^x - e^(x))/(e^x +e^(-x))+1`is invertible , find f-1.
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.
Which of the following graphs represents a one-one function?
Let A = {x ∈ R : −4 ≤ x ≤ 4 and x ≠ 0} and f : A → R be defined by \[f\left( x \right) = \frac{\left| x \right|}{x}\]Write the range of f.
Let \[f : \left[ - \frac{\pi}{2}, \frac{\pi}{2} \right] \to\] A be defined by f(x) = sin x. If f is a bijection, write set A.
If f(x) = x + 7 and g(x) = x − 7, x ∈ R, write fog (7).
Which one the following relations on A = {1, 2, 3} is a function?
f = {(1, 3), (2, 3), (3, 2)}, g = {(1, 2), (1, 3), (3, 1)} [NCERT EXEMPLAR]
If f(x) = 4 −( x - 7)3 then write f-1 (x).
Which of the following functions from
to itself are bijections?
The function
Which of the following functions from
\[A = \left\{ x \in R : - 1 \leq x \leq 1 \right\}\]
Let \[f\left( x \right) = \frac{1}{1 - x} . \text{Then}, \left\{ f o \left( fof \right) \right\} \left( x \right)\]
If the function
\[f : R \to R\] be such that
\[f\left( x \right) = x - \left[ x \right]\] where [x] denotes the greatest integer less than or equal to x, then \[f^{- 1} \left( x \right)\]
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.
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
g = {(1, 4), (2, 4), (3, 4)}
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 ______.
The function f : R → R defined by f(x) = 3 – 4x is ____________.
Let f : R `->` R be a function defined by f(x) = x3 + 4, then f is ______.
Given a function If as f(x) = 5x + 4, x ∈ R. If g : R → R is inverse of function ‘f then
Students of Grade 9, planned to plant saplings along straight lines, parallel to each other to one side of the playground ensuring that they had enough play area. Let us assume that they planted one of the rows of the saplings along the line y = x − 4. Let L be the set of all lines which are parallel on the ground and R be a relation on L.
Answer the following using the above information.
- The function f: R → R defined by f(x) = x − 4 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.
- Let f: R → R be defined by f(x) = x2 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.
- Let f: N → N be defined by f(x) = x2 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.
- Let : N → R be defined by f(x) = x2. Range of the function among the following is ____________.
Let [x] denote the greatest integer ≤ x, where x ∈ R. If the domain of the real valued function f(x) = `sqrt((|[x]| - 2)/(|[x]| - 3)` is (–∞, a) ∪ [b, c) ∪ [4, ∞), a < b < c, then the value of a + b + c is ______.
Let f: R→R be a polynomial function satisfying f(x + y) = f(x) + f(y) + 3xy(x + y) –1 ∀ x, y ∈ R and f'(0) = 1, then `lim_(x→∞)(f(2x))/(f(x)` is equal to ______.
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)
