Advertisements
Advertisements
Question
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
Advertisements
Solution
f : R → R is defined by f(x) = 3 – 4x
Let x1, x2 ∈ R such that f(x1) = f(x2)
⇒ 3 – 4x1 = 3 – 4x2
⇒ –4x1 = –4x2
⇒ x1 = x2
∴ f is one-one.
f : R → R be given for every y ∈ R (co-domain of f), there exists an element x ∈ R (domain of f) such that
f(x) = y
⇒ y = 3 – 4x
For any real number y ∈ R, there exists `x = (3 - y)/4 ∈ R` such that:
∴ `f((3 - y)/4)`
= `3 - 4 ((3 - y)/4)`
= 3 – 3 + y
= y
∴ f is onto.
Thus, f is one-one and onto and hence bijective.
APPEARS IN
RELATED QUESTIONS
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. Show that f is one-one.
Show that the function f : R → R given by f(x) = x3 is injective.
Give examples of two functions f: N → Z and g: Z → Z such that g o f is injective but gis not injective.
(Hint: Consider f(x) = x and g(x) =|x|)
Let S = {a, b, c} and T = {1, 2, 3}. Find F−1 of the following functions F from S to T, if it exists.
F = {(a, 2), (b, 1), (c, 1)}
Classify the following function as injection, surjection or bijection :
f : Z → Z, defined by f(x) = x2 + x
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = x3 + 1
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = sin2x + cos2x
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}
Show that the logarithmic function f : R0+ → R given by f (x) loga x ,a> 0 is a bijection.
Find gof and fog when f : R → R and g : R → R is defined by f(x) = 8x3 and g(x) = x1/3.
Let f : R → R and g : R → R be defined by f(x) = x + 1 and g (x) = x − 1. Show that fog = gof = IR.
Find fog and gof if : f(x)= x + 1, g (x) = 2x + 3 .
if `f (x) = sqrt(1-x)` and g(x) = `log_e` x are two real functions, then describe functions fog and gof.
State with reason whether the following functions have inverse:
h : {2, 3, 4, 5} → {7, 9, 11, 13} with h = {(2, 7), (3, 9), (4, 11), (5, 13)}
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 → (0, 2) defined by `f (x) =(e^x - e^(x))/(e^x +e^(-x))+1`is invertible , find f-1.
Let A and B be two sets, each with a finite number of elements. Assume that there is an injective map from A to B and that there is an injective map from B to A. Prove that there is a bijection from A to B.
Which one of the following graphs represents a function?
Let f : R → R+ be defined by f(x) = ax, a > 0 and a ≠ 1. Write f−1 (x).
Let `f : R - {- 3/5}` → R be a function defined as `f (x) = (2x)/(5x +3).`
f-1 : Range of f → `R -{-3/5}`.
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).
The function f : R → R defined by
`f (x) = 2^x + 2^(|x|)` is
The range of the function
\[f\left( x \right) =^{7 - x} P_{x - 3}\]
A function f from the set of natural numbers to integers defined by
`{([n-1]/2," when n is odd" is ),(-n/2,when n is even ) :}`
If \[g \left( f \left( x \right) \right) = \left| \sin x \right| \text{and} f \left( g \left( x \right) \right) = \left( \sin \sqrt{x} \right)^2 , \text{then}\]
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
Show that the function f: R → R defined by f(x) = `x/(x^2 + 1)`, ∀ ∈ + R , is neither one-one nor onto
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.
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 defined by f(x) = `1/x` ∀ x ∈ R. Then f is ______.
The number of bijective functions from set A to itself when A contains 106 elements is ____________.
The mapping f : N → N is given by f(n) = 1 + n2, n ∈ N when N is the set of natural numbers is ____________.
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.
- Ravi wants to find the number of injective functions from B to G. How many numbers of injective functions are possible?
If log102 = 0.3010.log103 = 0.4771 then the number of ciphers after decimal before a significant figure comes in `(5/3)^-100` is ______.
Difference between the greatest and least value of f(x) = `(1 + (cos^-1x)/π)^2 - (1 + (sin^-1x)/π)^2` is ______.
The graph of the function y = f(x) is symmetrical about the line x = 2, then ______.
