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 : Z → Z given by f(x) = x2
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.
In the following case, state whether the function is one-one, onto or bijective. Justify your answer.
f : R → R defined by f(x) = 3 – 4x
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 : N → N given by f(x) = x2
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 : Q − {3} → Q, defined by `f (x) = (2x +3)/(x-3)`
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}
Give examples of two one-one functions f1 and f2 from R to R, such that f1 + f2 : R → R. defined by (f1 + f2) (x) = f1 (x) + f2 (x) is not one-one.
Give examples of two surjective functions f1 and f2 from Z to Z such that f1 + f2 is not surjective.
Find gof and fog when f : R → R and g : R → R is defined by f(x) = 2x + 3 and g(x) = x2 + 5 .
Give examples of two functions f : N → Z and g : Z → Z, such that gof is injective but gis not injective.
Find fog and gof if : f (x) = |x|, g (x) = sin x .
Find fog and gof if : f(x)= x + 1, g (x) = 2x + 3 .
If f : R → (−1, 1) defined by `f (x) = (10^x- 10^-x)/(10^x + 10 ^-x)` is invertible, find f−1.
If f : A → A, g : A → A are two bijections, then prove that fog is an injection ?
If f : C → C is defined by f(x) = x4, write f−1 (1).
If f : R → R is defined by f(x) = x2, find f−1 (−25).
If f : R → R is defined by f(x) = 10 x − 7, then write f−1 (x).
If f(x) = x + 7 and g(x) = x − 7, x ∈ R, write fog (7).
Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. State whether f is one-one or not.
Let A = {a, b, c, d} and f : A → A be given by f = {( a,b ),( b , d ),( c , a ) , ( d , c )} write `f^-1`. [NCERT EXEMPLAR]
The range of the function
\[f\left( x \right) =^{7 - x} P_{x - 3}\]
Which of the following functions from
to itself are bijections?
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
Let N be the set of natural numbers and the function f: N → N be defined by f(n) = 2n + 3 ∀ n ∈ N. Then f is ______.
Let f: R → R be defined by f(x) = 3x – 4. Then f–1(x) is given by ______.
Let f: R → R be defined by f(x) = x2 + 1. Then, pre-images of 17 and – 3, respectively, are ______.
Let D be the domain of the real valued function f defined by f(x) = `sqrt(25 - x^2)`. Then, write D
If f: R → R is defined by f(x) = x2 – 3x + 2, write f(f (x))
Let g(x) = x2 – 4x – 5, then ____________.
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 ____________.
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: {1,2,3,....} → {1,4,9,....} be defined by f(x) = x2 is ____________.
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 f: R→Rbe defined as f (x) = `(x^2 + 1)/2`, then ______.
