Advertisements
Advertisements
प्रश्न
Show that if f1 and f2 are one-one maps from R to R, then the product f1 × f2 : R → R defined by (f1 × f2) (x) = f1 (x) f2 (x) need not be one - one.
Advertisements
उत्तर
We know that f1: R → R, given by f1(x) = x, and f2(x) = x are one-one.
Proving f1 is one-one:
Let x and y be two elements in the domain R, such that
f1(x) = f1(y)
⇒ x = y
So, f1 is one-one.
Proving f2 is one-one:
Let x and y be two elements in the domain R, such that
f2(x) = f2(y)
⇒ x = y
So, f2 is one-one.
Proving f1 × f2 is not one-one:
Given:
(f1 × f2) (x) = f1 (x) × f2 (x) = x × x = x2
Let x and y be two elements in the domain R, such that
( f1 × f2) (x) = (f1 × f2) (y)
⇒ x2 = y2
⇒ x = ± y
So, (f1 × f2) is not one-one.
APPEARS IN
संबंधित प्रश्न
Check the injectivity and surjectivity of the following function:
f : R → R given by f(x) = x2
Check the injectivity and surjectivity of the following function:
f : N → N given by f(x) = x3
Let f : R → R be defined as f(x) = 3x. Choose the correct answer.
Let f: R → R be the Signum Function defined as
f(x) = `{(1,x>0), (0, x =0),(-1, x< 0):}`
and g: R → R be the Greatest Integer Function given by g(x) = [x], where [x] is greatest integer less than or equal to x. Then does fog and gof coincide in (0, 1]?
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 : R → R, defined by f(x) = sinx
Find fog (2) and gof (1) when : f : R → R ; f(x) = x2 + 8 and g : R → R; g(x) = 3x3 + 1.
Find fog and gof if : f (x) = x2 g(x) = cos x .
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
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]
If f : A → A, g : A → A are two bijections, then prove that fog is a surjection ?
Which of the following graphs represents a one-one function?
Let `f : R - {- 3/5}` → R be a function defined as `f (x) = (2x)/(5x +3).`
f-1 : Range of f → `R -{-3/5}`.
Let A = {1, 2, 3, 4} and B = {a, b} be two sets. Write the total number of onto functions from A to B.
Write the domain of the real function f defined by f(x) = `sqrt (25 -x^2)` [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]
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
If \[f : R \to R is given by f\left( x \right) = 3x - 5, then f^{- 1} \left( x \right)\]
If \[g \left( f \left( x \right) \right) = \left| \sin x \right| \text{and} f \left( g \left( x \right) \right) = \left( \sin \sqrt{x} \right)^2 , \text{then}\]
Write about strlen() function.
Consider the set A containing n elements. Then, the total number of injective functions from A onto itself is ______
Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:
f(x) = `x/2`
Let f: `[2, oo)` → R be the function defined by f(x) = x2 – 4x + 5, then the range of f is ______.
The function f : A → B defined by f(x) = 4x + 7, x ∈ R is ____________.
Let A = R – {3}, B = R – {1}. Let f : A → B be defined by `"f"("x") = ("x" - 2)/("x" - 3)` Then, ____________.
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.
- Let f: R → R be defined by f(x) = x − 4. Then the range of f(x) is ____________.
A function f: x → y is/are called onto (or surjective) if x under f.
Let n(A) = 4 and n(B) = 6, Then the number of one – one functions from 'A' to 'B' is:
Difference between the greatest and least value of f(x) = `(1 + (cos^-1x)/π)^2 - (1 + (sin^-1x)/π)^2` is ______.
The function f(x) = [x], where [x] denotes the greatest integer less than or equal to x; is continuous at ______.
Let f(x) be a polynomial function of degree 6 such that `d/dx (f(x))` = (x – 1)3 (x – 3)2, then
Assertion (A): f(x) has a minimum at x = 1.
Reason (R): When `d/dx (f(x)) < 0, ∀ x ∈ (a - h, a)` and `d/dx (f(x)) > 0, ∀ x ∈ (a, a + h)`; where 'h' is an infinitesimally small positive quantity, then f(x) has a minimum at x = a, provided f(x) is continuous at x = a.
The given function f : R → R is not ‘onto’ function. Give reason.
Let f: R→Rbe defined as f (x) = `(x^2 + 1)/2`, then ______.
A function is called into if:
A function is called bijective if it is both:
If every seat in a classroom is occupied by one student, the situation resembles:
