Advertisements
Advertisements
प्रश्न
Find gof and fog when f : R → R and g : R → R is defined by f(x) = 2x + x2 and g(x) = x3
Advertisements
उत्तर
Given, f : R → R and g : R → R
So, gof : R → R and fog : R → R
f(x) = 2x + x2 and g(x) = x3
(gof) (x)
= g (f (x))
= g (2x+x2)
= (2x+x2)3
(fog) (x)
= f (g (x))
= f (x3)
= 2 (x3)+(x3)2
=2x3+x6
APPEARS IN
संबंधित प्रश्न
Show that the function f : R → R given by f(x) = x3 is injective.
Show that the function f: ℝ → ℝ defined by f(x) = `x/(x^2 + 1), ∀x in R`is neither one-one nor onto. Also, if g: ℝ → ℝ is defined as g(x) = 2x - 1. Find fog(x)
Classify the following function as injection, surjection or bijection : f : N → N given by f(x) = x3
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}
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.
Give examples of two functions f : N → Z and g : Z → Z, such that gof is injective but gis not injective.
Find fog and gof if : f (x) = x2 g(x) = cos x .
If f(x) = sin x and g(x) = 2x be two real functions, then describe gof and fog. Are these equal functions?
Consider f : R → R+ → [4, ∞) given by f(x) = x2 + 4. Show that f is invertible with inverse f−1 of f given by f−1 `(x)= sqrt (x-4)` where R+ is the set of all non-negative real numbers.
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).
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 = {x &epsis; R | −1 ≤ x ≤ 1} and let f : A → A, g : A → A be two functions defined by f(x) = x2 and g(x) = sin (π x/2). Show that g−1 exists but f−1 does not exist. Also, find g−1.
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.
Which one of the following graphs represents a function?
If A = {a, b, c} and B = {−2, −1, 0, 1, 2}, write the total number of one-one functions from A to B.
Let \[f : \left[ - \frac{\pi}{2}, \frac{\pi}{2} \right] \to\] A be defined by f(x) = sin x. If f is a bijection, write set A.
Let f : R → R, g : R → R be two functions defined by f(x) = x2 + x + 1 and g(x) = 1 − x2. Write fog (−2).
Which of the following functions from
\[A = \left\{ x \in R : - 1 \leq x \leq 1 \right\}\]
Let \[f\left( x \right) = \frac{1}{1 - x} . \text{Then}, \left\{ f o \left( fof \right) \right\} \left( x \right)\]
Let \[f\left( x \right) = \frac{\alpha x}{x + 1}, x \neq - 1\] Then, for what value of α is \[f \left( f\left( x \right) \right) = x?\]
Let the function f: R → R be defined by f(x) = 4x – 1, ∀ x ∈ R. Then, show that f is one-one.
Let A = {1, 2, 3, ...n} and B = {a, b}. Then the number of surjections from A into B is ______.
The function f : R → R defined by f(x) = 3 – 4x is ____________.
The function f : R → R given by f(x) = x3 – 1 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.
- 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.
- Let f: N → N be defined by f(x) = x2 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.
- Let : N → R be defined by f(x) = x2. Range of the function among the following is ____________.
If f: R → R given by f(x) =(3 − x3)1/3, find f0f(x)
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 a function `f: N rightarrow N` be defined by
f(n) = `{:[(2n",", n = 2"," 4"," 6"," 8","......),(n - 1",", n = 3"," 7"," 11"," 15","......),((n + 1)/2",", n = 1"," 5"," 9"," 13","......):}`
then f is ______.
Which one of the following graphs is a function of x?
 |
 |
| Graph A | Graph B |
The trigonometric equation tan–1x = 3tan–1 a has solution for ______.
A function is called bijective if it is both:
Which condition represents a one-one (injective) function?
Which condition represents an onto (surjective) function?
