Advertisements
Advertisements
प्रश्न
If f : A → B and g : B → C are one-one functions, show that gof is a one-one function.
Advertisements
उत्तर
Given, f : A → B and g : B → C are one - one.
Then, gof : A → B
Let us take two elements x and y from A, such that
(gof) (x) = (gof) (y)
⇒ g (f (x)) = g (f (y))
⇒ f (x) = f (y) (As, g is one-one)
⇒ x = y (As, f is one-one)
Hence, gof is one-one.
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 : N → N given by f(x) = x3
Show that the modulus function f : R → R given by f(x) = |x| is neither one-one nor onto, where |x| is x if x is positive or 0 and |x| is − x if x is negative.
Show that the Signum Function f : R → R, given by `f(x) = {(1", if" x > 0), (0", if" x = 0), (-1", if" x < 0):}` is neither one-one nor onto.
In the following case, state whether the function is one-one, onto or bijective. Justify your answer.
f : R → R defined by f(x) = 1 + x2
Prove that the function f : N → N, defined by f(x) = x2 + x + 1, is one-one but not onto
Let A = {−1, 0, 1} and f = {(x, x2) : x ∈ A}. Show that f : A → A is neither one-one nor onto.
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.
If f(x) = sin x and g(x) = 2x be two real functions, then describe gof and fog. Are these equal functions?
State with reason whether the following functions have inverse :
g : {5, 6, 7, 8} → {1, 2, 3, 4} with g = {(5, 4), (6, 3), (7, 4), (8, 2)}
Find f −1 if it exists : f : A → B, where A = {1, 3, 5, 7, 9}; B = {0, 1, 9, 25, 49, 81} and f(x) = x2
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)}.
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.
Let f be a function from C (set of all complex numbers) to itself given by f(x) = x3. Write f−1 (−1).
Let f : R − {−1} → R − {1} be given by\[f\left( x \right) = \frac{x}{x + 1} . \text{Write } f^{- 1} \left( x \right)\]
Write whether f : R → R, given by `f(x) = x + sqrtx^2` is one-one, many-one, onto or into.
Let f : R → R be the function defined by f(x) = 4x − 3 for all x ∈ R Then write f . [NCERT EXEMPLAR]
The function f : R → R defined by
`f (x) = 2^x + 2^(|x|)` is
The function \[f : [0, \infty ) \to \text {R given by } f\left( x \right) = \frac{x}{x + 1} is\]
Let
\[A = \left\{ x : - 1 \leq x \leq 1 \right\} \text{and} f : A \to \text{A such that f}\left( x \right) = x|x|\]
The function f : [-1/2, 1/2, 1/2] → [-π /2,π/2], defined by f (x) = `sin^-1` (3x - `4x^3`), is
The function \[f : R \to R\] defined by
\[f\left( x \right) = 6^x + 6^{|x|}\] is
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 \[f : R \to \left( - 1, 1 \right)\] is defined by
\[f\left( x \right) = \frac{- x|x|}{1 + x^2}, \text{ then } f^{- 1} \left( x \right)\] equals
Mark the correct alternative in the following question:
Let f : R \[-\] \[\left\{ \frac{3}{5} \right\}\] \[\to\] R be defined by f(x) = \[\frac{3x + 2}{5x - 3}\] Then,
Let the function f: R → R be defined by f(x) = 4x – 1, ∀ x ∈ R. Then, show that f is one-one.
Let f, g: R → R be two functions defined as f(x) = |x| + x and g(x) = x – x ∀ x ∈ R. Then, find f o g and g o f
Let A = {0, 1} and N be the set of natural numbers. Then the mapping f: N → A defined by f(2n – 1) = 0, f(2n) = 1, ∀ n ∈ N, is onto.
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
Let A = R – {3}, B = R – {1}. Let f : A → B be defined by `"f"("x") = ("x" - 2)/("x" - 3)` Then, ____________.
The domain of the function `"f"("x") = 1/(sqrt ({"sin x"} + {"sin" ( pi + "x")}))` where {.} denotes fractional part, is
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 ____________.
Let n(A) = 4 and n(B) = 6, Then the number of one – one functions from 'A' to 'B' is:
The domain of the function `cos^-1((2sin^-1(1/(4x^2-1)))/π)` 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 ______.
Difference between the greatest and least value of f(x) = `(1 + (cos^-1x)/π)^2 - (1 + (sin^-1x)/π)^2` is ______.
Let x is a real number such that are functions involved are well defined then the value of `lim_(t→0)[max{(sin^-1 x/3 + cos^-1 x/3)^2, min(x^2 + 4x + 7)}]((sin^-1t)/t)` where [.] is greatest integer function and all other brackets are usual brackets.
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 ______.
