Advertisements
Advertisements
प्रश्न
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = 3 − 4x
Advertisements
उत्तर
f : R → R, defined by f(x) = 3 − 4x
Injection test:
Let x and y be any two elements in the domain (R), such that f(x) = f(y).
f(x) = f(y)
3−4x = 3−4y
−4x = −4y
x = y
So, f is an injection .
Surjection test:
Let y be any element in the co-domain (R), such that f(x) = y for some element x in R (domain).
f(x) = y
3 − 4x = y
4x = 3−y
`x = (3-y)/4`∈ R
So, f is a surjection and f is a bijection.
APPEARS IN
संबंधित प्रश्न
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.
Let f: R → R be defined as f(x) = 10x + 7. Find the function g: R → R such that g o f = f o g = 1R.
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.
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)
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 : R → R, defined by f(x) = |x|
If f : A → B and g : B → C are one-one functions, show that gof is a one-one function.
Find fog and gof if : f(x)= x + 1, g (x) = 2x + 3 .
Let f(x) = x2 + x + 1 and g(x) = sin x. Show that fog ≠ gof.
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
Find f −1 if it exists : f : A → B, where A = {0, −1, −3, 2}; B = {−9, −3, 0, 6} and f(x) = 3 x.
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 `f : R - {- 3/5}` → R be a function defined as `f (x) = (2x)/(5x +3).`
f-1 : Range of f → `R -{-3/5}`.
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).
Let f be an invertible real function. Write ( f-1 of ) (1) + ( f-1 of ) (2) +..... +( f-1 of ) (100 )
If f : {5, 6} → {2, 3} and g : {2, 3} → {5, 6} are given by f = {(5, 2), (6, 3)} and g = {(2, 5), (3, 6)}, then find fog. [NCERT EXEMPLAR]
Let
\[A = \left\{ x \in R : - 1 \leq x \leq 1 \right\} = B\] Then, the mapping\[f : A \to \text{B given by} f\left( x \right) = x\left| x \right|\] is
If a function\[f : [2, \infty )\text{ to B defined by f}\left( x \right) = x^2 - 4x + 5\] is a bijection, then B =
The function f : [-1/2, 1/2, 1/2] → [-π /2,π/2], defined by f (x) = `sin^-1` (3x - `4x^3`), is
Let
\[f : R \to R\] be a function defined by
If \[F : [1, \infty ) \to [2, \infty )\] is given by
\[f\left( x \right) = x + \frac{1}{x}, then f^{- 1} \left( x \right)\]
If f(x) = `(x+3)/(4x−5) , "g"(x) = (3+5x)/(4x−1)` then verify that `("fog") (x)` = x.
Write about strcmp() function.
Let f: R → R be defined by f(x) = x2 + 1. Then, pre-images of 17 and – 3, respectively, are ______.
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 the function f: R → R be defined by f(x) = cosx, ∀ x ∈ R. Show that f is neither one-one nor onto
Let A = {1, 2, 3, ...n} and B = {a, b}. Then the number of surjections from A into B is ______.
Let f: R → R be defined by f(x) = `1/x` ∀ x ∈ R. Then f 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.
- Ravi wants to find the number of injective functions from B to G. How many numbers of injective functions are possible?
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 ____________.
If f: R→R is a function defined by f(x) = `[x - 1]cos((2x - 1)/2)π`, where [ ] denotes the greatest integer function, then f is ______.
If log102 = 0.3010.log103 = 0.4771 then the number of ciphers after decimal before a significant figure comes in `(5/3)^-100` 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 f: [0, 1]→[0, 1] is defined by f(x) = `(x + 1)/4` and `d/(dx) underbrace(((fofof......of)(x)))_("n" "times")""|_(x = 1/2) = 1/"m"^"n"`, m ∈ N, then the value of 'm' is ______.
The graph of the function y = f(x) is symmetrical about the line x = 2, then ______.
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 ______.
The trigonometric equation tan–1x = 3tan–1 a has solution for ______.
