Advertisements
Advertisements
Question
Let
\[A = \left\{ x \in R : x \leq 1 \right\} and f : A \to A\] be defined as
\[f\left( x \right) = x \left( 2 - x \right)\] Then,
\[f^{- 1} \left( x \right)\] is
Options
\[1 + \sqrt{1 - x}\]
\[1 - \sqrt{1 - x}\]
\[\sqrt{1 - x}\]
\[1 \pm \sqrt{1 - x}\]
Advertisements
Solution
LaTeX
\[\text{Let y be the element in the codomain R such that}\]
\[ f^{- 1} \left( x \right) = y . . . \left( 1 \right)\]
\[ \Rightarrow f\left( y \right) = x and y \leq1 \]
\[ \Rightarrow y\left( 2 - y \right) = x\]
\[ \Rightarrow 2y - y^2 = x\]
\[ \Rightarrow y^2 - 2y + x = 0\]
\[ \Rightarrow y^2 - 2y = - x\]
\[ \Rightarrow y^2 - 2y + 1 = 1 - x\]
\[ \Rightarrow \left( y - 1 \right)^2 = 1 - x\]
\[ \Rightarrow y - 1 = \pm \sqrt{1 - x}\]
\[ \Rightarrow y = 1 \pm \sqrt{1 - x}\]
\[ \Rightarrow y = 1 - \sqrt{1 - x} \left ( \because y \leq1 \right)\]
The correct answer is (b).
APPEARS IN
RELATED QUESTIONS
Check the injectivity and surjectivity of the following function:
f : R → R given by f(x) = x2
Let f : R → R be defined as f(x) = x4. Choose the correct answer.
Show that the function f : R → R given by f(x) = x3 is injective.
Let A = {−1, 0, 1, 2}, B = {−4, −2, 0, 2} and f, g : A → B be functions defined by f(x) = x2 − x, x ∈ A and g(x) = `2|x - 1/2|- 1`, x ∈ A. Are f and g equal?
Justify your answer. (Hint: One may note that two functions f : A → B and g : A → B such that f(a) = g(a) ∀ a ∈ A are called equal functions.)
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = sinx
Classify the following function as injection, surjection or bijection :
f : Q − {3} → Q, defined by `f (x) = (2x +3)/(x-3)`
Let A = [-1, 1]. Then, discuss whether the following function from A to itself is one-one, onto or bijective : `f (x) = x/2`
Show that f : R→ R, given by f(x) = x — [x], is neither one-one nor onto.
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 R+ be the set of all non-negative real numbers. If f : R+ → R+ and g : R+ → R+ are defined as `f(x)=x^2` and `g(x)=+sqrtx` , find fog and gof. Are they equal functions ?
Find fog and gof if : f (x) = x+1, g (x) = sin x .
Consider f : R → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f.
Consider f : R+ → [−5, ∞) given by f(x) = 9x2 + 6x − 5. Show that f is invertible with `f^-1 (x) = (sqrt (x +6)-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.
If f : A → A, g : A → A are two bijections, then prove that fog is a surjection ?
Which one of the following graphs represents a function?
Which of the following graphs represents a one-one function?
If A = {1, 2, 3} and B = {a, b}, write the total number of functions from A to B.
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 : R → R be defined as `f (x) = (2x - 3)/4.` write fo f-1 (1) .
Let A = {1, 2, 3, 4} and B = {a, b} be two sets. Write the total number of onto functions from A to B.
The range of the function
\[f\left( x \right) =^{7 - x} P_{x - 3}\]
Let
\[f : R - \left\{ n \right\} \to R\]
Which of the following functions from
\[A = \left\{ x \in R : - 1 \leq x \leq 1 \right\}\]
If \[f : R \to R is given by f\left( x \right) = 3x - 5, then f^{- 1} \left( x \right)\]
If \[f\left( x \right) = \sin^2 x\] and the composite function \[g\left( f\left( x \right) \right) = \left| \sin x \right|\] then g(x) is equal to
Mark the correct alternative in the following question:
Let f : R → R be given by f(x) = tanx. Then, f-1(1) is
A function f: R→ R defined by f(x) = `(3x) /5 + 2`, x ∈ R. Show that f is one-one and onto. Hence find f−1.
Write about strlen() function.
Let f: R → R be the function defined by f(x) = 4x – 3 ∀ x ∈ R. Then write f–1
Consider the set A containing n elements. Then, the total number of injective functions from A onto itself is ______
Let D be the domain of the real valued function f defined by f(x) = `sqrt(25 - x^2)`. Then, write D
Let g(x) = x2 – 4x – 5, then ____________.
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}
- Three friends F1, F2, and F3 exercised their voting right in general election-2019, then which of the following is true?
Students of Grade 9, planned to plant saplings along straight lines, parallel to each other to one side of the playground ensuring that they had enough play area. Let us assume that they planted one of the rows of the saplings along the line y = x − 4. Let L be the set of all lines which are parallel on the ground and R be a relation on L.
Answer the following using the above information.
- Let f: R → R be defined by f(x) = x − 4. Then the range of f(x) is ____________.
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 ______.
The solution set of the inequation log1/3(x2 + x + 1) + 1 > 0 is ______.
Write the domain and range (principle value branch) of the following functions:
f(x) = tan–1 x.
