Advertisements
Advertisements
Question
Let A = {a, b, c}, B = {u v, w} and let f and g be two functions from A to B and from B to A, respectively, defined as :
f = {(a, v), (b, u), (c, w)}, g = {(u, b), (v, a), (w, c)}.
Show that f and g both are bijections and find fog and gof.
Advertisements
Solution
Proving f is a bijection :
f = {(a, v), (b, u), (c, w)} and f : A → B
Injectivity of f: No two elements of A have the same image in B.
So, f is one-one.
Surjectivity of f: Co-domain of f = {u v, w}
Range of f = {u v, w}
Both are same.
So, f is onto.
Hence, f is a bijection.
Proving g is a bijection :
g = {(u, b), (v, a), (w, c)} and g : B → A
Injectivity of g: No two elements of B have the same image in A.
So, g is one-one.
Surjectivity of g: Co-domain of g = {a, b, c}
Range of g = {a, b, c}
Both are the same.
So, g is onto.
Hence, g is a bijection.
Finding fog :
Co-domain of g is same as the domain of f.
So, fog exists and fog : {u v, w} → {u v, w}
(fog) (u) = f (g (u)) = f (b) = u
(fog) (v) = f (g (v)) = f (a) = v
(fog) (w) = f (g (w)) = f (c) = w
So, fog = { (u, u), (v, v), (w, w) }
Finding gof :
Co-domain of f is same as the domain of g.
So, fog exists and gof : {a, b, c} → {a, b, c}
(gof) (a) = g (f (a)) = g (v) = a
(gof) (b) = g (f (b)) = g (u) = b
(gof) (c) = g (f (c)) = g (w) = c
So, gof = { (a, a), (b, b), (c, c) }
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.
Let A = R – {3} and B = R – {1}. Consider the function f : A → B defined by f(x) = `((x - 2)/(x - 3))`. Is f one-one and onto? Justify your answer.
Show that the function f : R → {x ∈ R : –1 < x < 1} defined by f(x) = `x/(1 + |x|)`, x ∈ R is one-one and onto function.
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|)
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, defined by f(x) = x2 + x
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}
Find the number of all onto functions from the set A = {1, 2, 3, ..., n} to itself.
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.
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.
Find gof and fog when f : R → R and g : R → R is defined by f(x) = x and g(x) = |x| .
Find fog (2) and gof (1) when : f : R → R ; f(x) = x2 + 8 and g : R → R; g(x) = 3x3 + 1.
If f : A → B and g : B → C are one-one functions, show that gof is a one-one function.
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]
Let f : [−1, ∞) → [−1, ∞) be given by f(x) = (x + 1)2 − 1, x ≥ −1. Show that f is invertible. Also, find the set S = {x : f(x) = f−1 (x)}.
Let f be an invertible real function. Write ( f-1 of ) (1) + ( f-1 of ) (2) +..... +( f-1 of ) (100 )
What is the range of the function
`f (x) = ([x - 1])/(x -1) ?`
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]
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]
If f(x) = 4 −( x - 7)3 then write f-1 (x).
The range of the function
\[f\left( x \right) =^{7 - x} P_{x - 3}\]
If a function\[f : [2, \infty )\text{ to B defined by f}\left( x \right) = x^2 - 4x + 5\] is a bijection, then B =
Let
\[f : R \to R\] be a function defined by
Let
\[f : [2, \infty ) \to X\] be defined by
\[f\left( x \right) = 4x - x^2\] Then, f is invertible if X =
Let the function f: R → R be defined by f(x) = 4x – 1, ∀ x ∈ R. Then, show that f is one-one.
Let f: R → R be the function defined by f(x) = 4x – 3 ∀ x ∈ R. Then write f–1
For sets A, B and C, let f: A → B, g: B → C be functions such that g o f is injective. Then both f and g are injective functions.
Are the following set of ordered pairs functions? If so, examine whether the mapping is injective or surjective.
{(a, b): a is a person, b is an ancestor of a}
Let A = {1, 2, 3, ...n} and B = {a, b}. Then the number of surjections from A into B is ______.
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 ____________.
Given a function If as f(x) = 5x + 4, x ∈ R. If g : R → R is inverse of function ‘f then
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 ____________.
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
Let f: R→R be a continuous function such that f(x) + f(x + 1) = 2, for all x ∈ R. If I1 = `int_0^8f(x)dx` and I2 = `int_(-1)^3f(x)dx`, then the value of I1 + 2I2 is equal to ______.
Let f(x) be a polynomial of degree 3 such that f(k) = `-2/k` for k = 2, 3, 4, 5. Then the value of 52 – 10f(10) is equal to ______.
Let S = {1, 2, 3, 4, 5, 6, 7}. Then the number of possible functions f: S `rightarrow` S such that f(m.n) = f(m).f(n) for every m, n ∈ S and m.n ∈ S is equal to ______.
If f : R `rightarrow` R is defined by `f(x) = (2x - 7)/4`, show that f(x) is one-one and onto.
