Advertisements
Advertisements
प्रश्न
If a function\[f : [2, \infty )\text{ to B defined by f}\left( x \right) = x^2 - 4x + 5\] is a bijection, then B =
पर्याय
R
[1, ∞)
[4, ∞)
[5, ∞)
Advertisements
उत्तर
Since f is a bijection, co-domain of f = range of f
⇒ B = range of f
\[\text{Given}: f\left( x \right) = x^2 - 4x + 5\]
\[\text{Let}f\left( x \right) = y\]
\[ \Rightarrow y = x^2 - 4x + 5\]
\[ \Rightarrow x^2 - 4x + \left( 5 - y \right) = 0\]
\[ \because \text{Discrimant}, D = b^2 - 4ac \geq 0, \]
\[ \left( - 4 \right)^2 - 4 \times 1 \times \left( 5 - y \right) \geq 0\]
\[ \Rightarrow 16 - 20 + 4y \geq 0\]
\[ \Rightarrow 4y \geq 4\]
\[ \Rightarrow y \geq 1\]
\[ \Rightarrow y \in [1, \infty )\]
\[ \Rightarrow \text{Range of f} = [1, \infty )\]
\[ \Rightarrow B = [1, \infty )\]
So, the answer is (b).
APPEARS IN
संबंधित प्रश्न
Check the injectivity and surjectivity of the following function:
f : Z → Z 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.
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
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.
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, 2), (b, 1), (c, 1)}
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
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = x3 + 1
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = 1 + x2
Show that the logarithmic function f : R0+ → R given by f (x) loga x ,a> 0 is a bijection.
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) = x and g(x) = |x| .
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.
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.
If f(x) = sin x and g(x) = 2x be two real functions, then describe gof and fog. Are these equal functions?
if f (x) = `sqrt (x +3) and g (x) = x ^2 + 1` be two real functions, then find fog and gof.
State with reason whether the following functions have inverse :
f : {1, 2, 3, 4} → {10} with f = {(1, 10), (2, 10), (3, 10), (4, 10)}
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.
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 = {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 f : R → R is defined by f(x) = x2, find f−1 (−25).
The function f : [-1/2, 1/2, 1/2] → [-π /2,π/2], defined by f (x) = `sin^-1` (3x - `4x^3`), is
Let \[f\left( x \right) = \frac{1}{1 - x} . \text{Then}, \left\{ f o \left( fof \right) \right\} \left( x \right)\]
If the function
\[f : R \to R\] be such that
\[f\left( x \right) = x - \left[ x \right]\] where [x] denotes the greatest integer less than or equal to x, then \[f^{- 1} \left( x \right)\]
Let f: R → R be the function defined by f(x) = 4x – 3 ∀ x ∈ R. Then write f–1
Let A be a finite set. Then, each injective function from A into itself is not surjective.
Let f: R → R be the function defined by f(x) = 2x – 3 ∀ x ∈ R. write f–1
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: R → R be given by f(x) = tan x. Then f–1(1) is ______.
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.
The function f : A → B defined by f(x) = 4x + 7, x ∈ R is ____________.
Let X = {-1, 0, 1}, Y = {0, 2} and a function f : X → Y defiend by y = 2x4, is ____________.
Let f : [0, ∞) → [0, 2] be defined by `"f" ("x") = (2"x")/(1 + "x"),` then f 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 know among those relations, how many functions can be formed from B to G?
Let n(A) = 4 and n(B) = 6, Then the number of one – one functions from 'A' to 'B' is:
Let f: R→R be defined as f(x) = 2x – 1 and g: R – {1}→R be defined as g(x) = `(x - 1/2)/(x - 1)`. Then the composition function f (g(x)) is ______.
