Advertisements
Advertisements
प्रश्न
Let \[f\left( x \right) = x^2 and g\left( x \right) = 2^x\] Then, the solution set of the equation
विकल्प
R
{0}
{0, 2}
none of these
Advertisements
उत्तर
\[\text{Since}\left( \text{fog} \right)\left( x \right) = \left( \text{gof} \right)\left( x \right), \]
\[ f\left( g\left( x \right) \right) = g\left( f\left( x \right) \right)\]
\[ \Rightarrow f\left( 2^x \right) = g\left( x^2 \right)\]
\[ \Rightarrow \left( 2^x \right)^2 = 2^{x^2} \]
\[ \Rightarrow 2^{2x} = 2^{x^2} \]
\[ \Rightarrow x^2 = 2x\]
\[ \Rightarrow x^2 - 2x = 0\]
\[ \Rightarrow x\left( x - 2 \right) = 0\]
\[ \Rightarrow x = 0, 2\]
\[ \Rightarrow x \in \left\{ 0, 2 \right\}\]
So, the answer is (c) .
APPEARS IN
संबंधित प्रश्न
Check the injectivity and surjectivity of the following function:
f : R → R given by f(x) = x2
Which of the following functions from A to B are one-one and onto?
f2 = {(2, a), (3, b), (4, c)} ; A = {2, 3, 4}, B = {a, b, c}
Show that the function f : R − {3} → R − {2} given by f(x) = `(x-2)/(x-3)` is a bijection.
Let A = [-1, 1]. Then, discuss whether the following function from A to itself is one-one, onto or bijective : `f (x) = x/2`
Set of ordered pair of a function? If so, examine whether the mapping is injective or surjective :{(x, y) : x is a person, y is the mother of x}
Suppose f1 and f2 are non-zero one-one functions from R to R. Is `f_1 / f^2` necessarily one - one? Justify your answer. Here,`f_1/f_2 : R → R is given by (f_1/f_2) (x) = (f_1(x))/(f_2 (x)) for all x in R .`
Show that f : R→ R, given by f(x) = x — [x], is neither one-one nor onto.
Let f = {(3, 1), (9, 3), (12, 4)} and g = {(1, 3), (3, 3) (4, 9) (5, 9)}. Show that gof and fog are both defined. Also, find fog and gof.
Find fog and gof if : f(x) = sin−1 x, g(x) = x2
State with reason whether the following functions have inverse:
h : {2, 3, 4, 5} → {7, 9, 11, 13} with h = {(2, 7), (3, 9), (4, 11), (5, 13)}
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.
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 f : R → (−1, 1) defined by `f (x) = (10^x- 10^-x)/(10^x + 10 ^-x)` 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.
Let f be a function from R to R, such that f(x) = cos (x + 2). Is f invertible? Justify your answer.
Which one of the following graphs represents a function?
If f : R → R is given by f(x) = x3, write f−1 (1).
Let f : R − {−1} → R − {1} be given by\[f\left( x \right) = \frac{x}{x + 1} . \text{Write } f^{- 1} \left( x \right)\]
Write the domain of the real function
`f (x) = sqrt([x] - x) .`
Let the function
\[f : R - \left\{ - b \right\} \to R - \left\{ 1 \right\}\]
\[f\left( x \right) = \frac{x + a}{x + b}, a \neq b .\text{Then},\]
The function \[f : [0, \infty ) \to \text {R given by } f\left( x \right) = \frac{x}{x + 1} is\]
Which of the following functions form Z to itself are bijections?
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
Let
\[f : [2, \infty ) \to X\] be defined by
\[f\left( x \right) = 4x - x^2\] Then, f is invertible if X =
Let A = ℝ − {3}, B = ℝ − {1}. Let f : A → B be defined by \[f\left( x \right) = \frac{x - 2}{x - 3}, \forall x \in A\] Show that f is bijective. Also, find
(i) x, if f−1(x) = 4
(ii) f−1(7)
Let f: R → R be the function defined by f(x) = 2x – 3 ∀ x ∈ R. write f–1
Let X = {1, 2, 3}and Y = {4, 5}. Find whether the following subset of X ×Y are function from X to Y or not
f = {(1, 4), (1, 5), (2, 4), (3, 5)}
Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:
g(x) = |x|
The function f : A → B defined by f(x) = 4x + 7, x ∈ R is ____________.
The number of bijective functions from set A to itself when A contains 106 elements is ____________.
The function f : R → R given by f(x) = x3 – 1 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?
If `f : R -> R^+ U {0}` be defined by `f(x) = x^2, x ∈ R`. The mapping is
'If 'f' is a linear function satisfying f[x + f(x)] = x + f(x), then f(5) can be equal to:
For x ∈ R, x ≠ 0, let f0(x) = `1/(1 - x)` and fn+1 (x) = f0(fn(x)), n = 0, 1, 2, .... Then the value of `f_100(3) + f_1(2/3) + f_2(3/2)` is equal to ______.
Write the domain and range (principle value branch) of the following functions:
f(x) = tan–1 x.
The trigonometric equation tan–1x = 3tan–1 a has solution for ______.
A function is a rule that assigns each element of one set to exactly how many elements of another set?
