Advertisements
Advertisements
प्रश्न
Let f(x) = x2 + x + 1 and g(x) = sin x. Show that fog ≠ gof.
Advertisements
उत्तर
(fog) (x) = f (g (x)) = f(sin x) = sin2 x + sin x + 1
and (gof) (x) = g (f (x)) = g (x2+ x + 1) = sin ( x2+ x + 1)
So, fog ≠ gof.
APPEARS IN
संबंधित प्रश्न
Check the injectivity and surjectivity of the following function:
f : R → R given by f(x) = x2
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.
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.
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, 3), (b, 2), (c, 1)}
Let f = {(1, −1), (4, −2), (9, −3), (16, 4)} and g = {(−1, −2), (−2, −4), (−3, −6), (4, 8)}. Show that gof is defined while fog is not defined. Also, find gof.
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.
Find fog and gof if : f (x) = ex g(x) = loge 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
if `f (x) = sqrt(1-x)` and g(x) = `log_e` x are two real functions, then describe functions fog and gof.
Show that the function f : Q → Q, defined by f(x) = 3x + 5, is invertible. Also, find f−1
A function f : R → R is defined as f(x) = x3 + 4. Is it a bijection or not? In case it is a bijection, find f−1 (3).
Let A = R - {3} and B = R - {1}. Consider the function f : A → B defined by f(x) = `(x-2)/(x-3).`Show that f is one-one and onto and hence find f-1.
[CBSE 2012, 2014]
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.
Let C denote the set of all complex numbers. A function f : C → C is defined by f(x) = x3. Write f−1(1).
If f : R → R is defined by f(x) = 10 x − 7, then write f−1 (x).
If f(x) = x + 7 and g(x) = x − 7, x ∈ R, write fog (7).
Let f : R → R be the function defined by f(x) = 4x − 3 for all x ∈ R Then write f . [NCERT EXEMPLAR]
\[f : A \to \text{B given by } 3^{ f\left( x \right)} + 2^{- x} = 4\] is a bijection, then
Let
\[A = \left\{ x \in R : x \geq 1 \right\}\] The inverse of the function,
\[f : A \to A\] given by
\[f\left( x \right) = 2^{x \left( x - 1 \right)} , is\]
The distinct linear functions that map [−1, 1] onto [0, 2] are
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 A = {1, 2, ... , n} and B = {a, b}. Then the number of subjections from A into B is
If f(x) = `(x+3)/(4x−5) , "g"(x) = (3+5x)/(4x−1)` then verify that `("fog") (x)` = x.
Write about strcmp() function.
The domain of the function f: R → R defined by f(x) = `sqrt(x^2 - 3x + 2)` is ______
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.
Let D be the domain of the real valued function f defined by f(x) = `sqrt(25 - x^2)`. Then, write D
Let f: R → R be the function defined by f(x) = 2x – 3 ∀ x ∈ R. write f–1
Let X = {1, 2, 3}and Y = {4, 5}. Find whether the following subset of X ×Y are function from X to Y or not
g = {(1, 4), (2, 4), (3, 4)}
Let A = R – {3}, B = R – {1}. Let f: A → B be defined by f(x) = `(x - 2)/(x - 3)` ∀ x ∈ A . Then show that f is bijective.
If the set A contains 5 elements and the set B contains 6 elements, then the number of one-one and onto mappings from A to B is ______.
Let f : [0, ∞) → [0, 2] be defined by `"f" ("x") = (2"x")/(1 + "x"),` then f 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.
- Ravi wants to find the number of injective functions from B to G. How many numbers of injective functions are possible?
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.
Let f: R→R be a polynomial function satisfying f(x + y) = f(x) + f(y) + 3xy(x + y) –1 ∀ x, y ∈ R and f'(0) = 1, then `lim_(x→∞)(f(2x))/(f(x)` is equal to ______.
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 ______.
Find the domain of sin–1 (x2 – 4).
If f : R `rightarrow` R is defined by `f(x) = (2x - 7)/4`, show that f(x) is one-one and onto.
