Advertisements
Advertisements
प्रश्न
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]
Advertisements
उत्तर
We have,
f : {1, 3, 4}
→ {1, 2, 5} and g : {1, 2, 5}
→ {1, 3}, are given by f = {(1, 2), (3, 5), (4, 1)} and g = {(2, 3), (5, 1), (1, 3)}, respectively
As,
\[fog\left( 2 \right) = f\left( g\left( 2 \right) \right) = f\left( 3 \right) = 5, \]
\[fog\left( 5 \right) = f\left( g\left( 5 \right) \right) = f\left( 1 \right) = 2, \]
\[fog\left( 1 \right) = f\left( g\left( 1 \right) \right) = f\left( 3 \right) = 5, \]
\[So, \]
\[fog : \left\{ 1, 2, 5 \right\} \to \left\{ 1, 2, 5 \right\} \text{ is given by}\]
\[fog = \left\{ \left( 2, 5 \right), \left( 5, 2 \right), \left( 1, 5 \right) \right\}\]
APPEARS IN
संबंधित प्रश्न
Check the injectivity and surjectivity of the following function:
f : N → N given by f(x) = x2
Check the injectivity and surjectivity of the following function:
f : R → R given by f(x) = x2
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|)
Give an example of a function which is one-one but not onto ?
Give an example of a function which is not one-one but onto ?
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = `x/(x^2 +1)`
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.
Let f : N → N be defined by
`f(n) = { (n+ 1, if n is odd),( n-1 , if n is even):}`
Show that f is a bijection.
[CBSE 2012, NCERT]
Find gof and fog when f : R → R and g : R → R is defined by f(x) = 2x + 3 and g(x) = x2 + 5 .
Let R+ be the set of all non-negative real numbers. If f : R+ → R+ and g : R+ → R+ are defined as `f(x)=x^2` and `g(x)=+sqrtx` , find fog and gof. Are they equal functions ?
If f : A → B and g : B → C are onto functions, show that gof is a onto function.
Let f(x) = x2 + x + 1 and g(x) = sin x. Show that fog ≠ gof.
Let
f (x) =`{ (1 + x, 0≤ x ≤ 2) , (3 -x , 2 < x ≤ 3):}`
Find fof.
Consider f : {1, 2, 3} → {a, b, c} and g : {a, b, c} → {apple, ball, cat} defined as f (1) = a, f (2) = b, f (3) = c, g (a) = apple, g (b) = ball and g (c) = cat. Show that f, g and gof are invertible. Find f−1, g−1 and gof−1and show that (gof)−1 = f −1o g−1
Which of the following graphs represents a one-one function?
If f : R → R is defined by f(x) = x2, write f−1 (25)
If f : R → R is defined by f(x) = x2, find f−1 (−25).
Let f be an invertible real function. Write ( f-1 of ) (1) + ( f-1 of ) (2) +..... +( f-1 of ) (100 )
Write the domain of the real function
`f (x) = 1/(sqrt([x] - x)`.
The function
f : A → B defined by
f (x) = - x2 + 6x - 8 is a bijection if
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
Let
Let \[f\left(x\right) = x^3\] be a function with domain {0, 1, 2, 3}. Then domain of \[f^{-1}\] is ______.
Write about strlen() function.
Let the function f: R → R be defined by f(x) = 4x – 1, ∀ x ∈ R. Then, show that f is one-one.
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
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 C be the set of complex numbers. Prove that the mapping f: C → R given by f(z) = |z|, ∀ z ∈ C, is neither one-one nor onto.
Let f: R → R be the functions defined by f(x) = x3 + 5. Then f–1(x) is ______.
Let f: `[2, oo)` → R be the function defined by f(x) = x2 – 4x + 5, then the range of f is ______.
Which of the following functions from Z into Z is bijective?
Let f : R → R be a function defined by f(x) `= ("e"^abs"x" - "e"^-"x")/("e"^"x" + "e"^-"x")` then f(x) is
Range of `"f"("x") = sqrt((1 - "cos x") sqrt ((1 - "cos x")sqrt ((1 - "cos x")....infty))`
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 ____________.
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(x) = ax (a > 0) be written as f(x) = f1(x) + f2(x), where f1(x) is an even function and f2(x) is an odd function. Then f1(x + y) + f1(x – y) equals ______.
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 ______.
