Advertisements
Advertisements
प्रश्न
Let
\[f : [2, \infty ) \to X\] be defined by
\[f\left( x \right) = 4x - x^2\] Then, f is invertible if X =
पर्याय
\[[2, \infty )\]
\[( - \infty , 2]\]
\[( - \infty , 4]\]
\[[4, \infty )\]
Advertisements
उत्तर
Since f is invertible, range of f = co domain of f = X
So, we need to find the range of f to find X.
For finding the range, let
\[f\left( x \right) = y\]
\[ \Rightarrow 4x - x^2 = y\]
\[ \Rightarrow x^2 - 4x = - y\]
\[ \Rightarrow x^2 - 4x + 4 = 4 - y\]
\[ \Rightarrow \left( x - 2 \right)^2 = 4 - y\]
\[ \Rightarrow x - 2 = \pm \sqrt{4 - y}\]
\[ \Rightarrow x = 2 \pm \sqrt{4 - y}\]
\[\text{This is defined only when}\]
\[4 - y \geq 0\]
\[ \Rightarrow y \leq 4\]
\[X = \text{Range of f} = ( - \infty , 4]\]
So, the answer is (c).
APPEARS IN
संबंधित प्रश्न
Show that the function f : R* → R* defined by f(x) = `1/x` is one-one and onto, where R* is the set of all non-zero real numbers. Is the result true, if the domain R* is replaced by N with co-domain being same as R?
Check the injectivity and surjectivity of the following function:
f : Z → Z given by f(x) = x2
If the function `f(x) = sqrt(2x - 3)` is invertible then find its inverse. Hence prove that `(fof^(-1))(x) = x`
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = x3 − x
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}
Show that the exponential function f : R → R, given by f(x) = ex, is one-one but not onto. What happens if the co-domain is replaced by`R0^+` (set of all positive real numbers)?
Let f : R → R and g : R → R be defined by f(x) = x + 1 and g (x) = x − 1. Show that fog = gof = IR.
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.
Let f, g, h be real functions given by f(x) = sin x, g (x) = 2x and h (x) = cos x. Prove that fog = go (fh).
Let
f (x) =`{ (1 + x, 0≤ x ≤ 2) , (3 -x , 2 < x ≤ 3):}`
Find fof.
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)}
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 `{- 4/3} `- 43 →">→ R be a function defined as f(x) = `(4x)/(3x +4)` . Show that f : R - `{-4/3}`→ Rang (f) is one-one and onto. Hence, find f -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.
If f : R → R is defined by f(x) = x2, find f−1 (−25).
If f : R → R, g : R → are given by f(x) = (x + 1)2 and g(x) = x2 + 1, then write the value of fog (−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 A = {1, 2, 3, 4} and B = {a, b} be two sets. Write the total number of onto functions from A to B.
Let M be the set of all 2 × 2 matrices with entries from the set R of real numbers. Then, the function f : M→ R defined by f(A) = |A| for every A ∈ M, is
A function f from the set of natural numbers to integers defined by
`{([n-1]/2," when n is odd" is ),(-n/2,when n is even ) :}`
Which of the following functions from
\[A = \left\{ x \in R : - 1 \leq x \leq 1 \right\}\]
Let A = R − (2) and B = R − (1). If f: A ⟶ B is a function defined by`"f(x)"=("x"-1)/("x"-2),` how that f is one-one and onto. Hence, find f−1.
Let D be the domain of the real valued function f defined by f(x) = `sqrt(25 - x^2)`. Then, write D
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.
The smallest integer function f(x) = [x] is ____________.
Which of the following functions from Z into Z is bijective?
Let f : R `->` R be a function defined by f(x) = x3 + 4, then f is ______.
A general election of Lok Sabha is a gigantic exercise. About 911 million people were eligible to vote and voter turnout was about 67%, the highest ever
Let I be the set of all citizens of India who were eligible to exercise their voting right in the general election held in 2019. A relation ‘R’ is defined on I as follows:
R = {(V1, V2) ∶ V1, V2 ∈ I and both use their voting right in the general election - 2019}
- Mr. ’X’ and his wife ‘W’ both exercised their voting right in the general election-2019, Which of the following is true?
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: R → R 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 f: N → N be defined by f(x) = x2 is ____________.
Let f: R → R defined by f(x) = x4. Choose the correct answer
A function f: x → y is said to be one – one (or injective) if:
A function f: x → y is/are called onto (or surjective) if x under f.
The domain of the function `cos^-1((2sin^-1(1/(4x^2-1)))/π)` is ______.
`x^(log_5x) > 5` implies ______.
Let a and b are two positive integers such that b ≠ 1. Let g(a, b) = Number of lattice points inside the quadrilateral formed by lines x = 0, y = 0, x = b and y = a. f(a, b) = `[a/b] + [(2a)/b] + ... + [((b - 1)a)/b]`, then the value of `[(g(101, 37))/(f(101, 37))]` is ______.
(Note P(x, y) is lattice point if x, y ∈ I)
(where [.] denotes greatest integer function)
A function f : [– 4, 4] `rightarrow` [0, 4] is given by f(x) = `sqrt(16 - x^2)`. Show that f is an onto function but not a one-one function. Further, find all possible values of 'a' for which f(a) = `sqrt(7)`.
