Advertisements
Advertisements
Question
if `f (x) = sqrt(1-x)` and g(x) = `log_e` x are two real functions, then describe functions fog and gof.
Advertisements
Solution
`f (x) = sqrt(1-x)`
For domain, 1-x≥0
⇒ x≤1
⇒ domain of f = (−∞, 1]
⇒ f : (−∞, 1] → (0,∞)
g(x) = loge x
Clearly, g : (0, ∞) → R
Computation of fog :
Clearly, the range of g is not a subset of the domain of f.
So,we need to compute the domain of fog.
⇒ Domain (fog) = {x : x ∈ Domain (g) and g(x) ∈ Domain of f}
⇒ Domain (fog) = {x: x ∈ (0, ∞) and loge x ∈ (−∞, 1]}
⇒ Domain (fog) = { x: x ∈ (0, ∞) and x ∈ (0, e] }
⇒ Domain (fog)= {x : x ∈ (0, e]}
⇒ Domain (fog )= (0, e]
⇒ fog : (0, e) → R
So, (fog) (x) = f (g (x))
= f (loge x)
= `sqrt( 1-log_e x)`
Computation of gof:Clearly, the range of f is a subset of the domain of g.
⇒ gof : (−∞,1] → R
⇒ (gof) (x) = g (f (x))
= `g (sqrt(1-x))`
= `log_e sqrt (1 - x)`
= `log_e (1 - x)^(1/2)`
= `1/2 log_e (1-x)`
APPEARS IN
RELATED QUESTIONS
If the function `f(x) = sqrt(2x - 3)` is invertible then find its inverse. Hence prove that `(fof^(-1))(x) = x`
Let A = {−1, 0, 1} and f = {(x, x2) : x ∈ A}. Show that f : A → A is neither one-one nor onto.
Classify the following function as injection, surjection or bijection :
f : Q → Q, defined by f(x) = x3 + 1
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = `x/(x^2 +1)`
Show that the logarithmic function f : R0+ → R given by f (x) loga x ,a> 0 is a bijection.
Find the number of all onto functions from the set A = {1, 2, 3, ..., n} to itself.
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 .`
Find fog and gof if : f (x) = x2 g(x) = cos x .
Find fog and gof if : f(x) = sin−1 x, g(x) = x2
Let
f (x) =`{ (1 + x, 0≤ x ≤ 2) , (3 -x , 2 < x ≤ 3):}`
Find fof.
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.
If f : A → A, g : A → A are two bijections, then prove that fog is a surjection ?
Let f be a function from C (set of all complex numbers) to itself given by f(x) = x3. Write f−1 (−1).
If f : C → C is defined by f(x) = x4, write f−1 (1).
If f : R → R is defined by f(x) = 10 x − 7, then write f−1 (x).
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 whether f : R → R, given by `f(x) = x + sqrtx^2` is one-one, many-one, onto or into.
Write the domain of the real function f defined by f(x) = `sqrt (25 -x^2)` [NCERT EXEMPLAR]
Let the function
\[f : R - \left\{ - b \right\} \to R - \left\{ 1 \right\}\]
\[f\left( x \right) = \frac{x + a}{x + b}, a \neq b .\text{Then},\]
Let
\[A = \left\{ x \in R : - 1 \leq x \leq 1 \right\} = B\] Then, the mapping\[f : A \to \text{B given by} f\left( x \right) = x\left| x \right|\] is
The function
\[f : R \to R\] defined by\[f\left( x \right) = \left( x - 1 \right) \left( x - 2 \right) \left( x - 3 \right)\]
(a) one-one but not onto
(b) onto but not one-one
(c) both one and onto
(d) neither one-one nor onto
Let
Let \[f\left( x \right) = x^2 and g\left( x \right) = 2^x\] Then, the solution set of the equation
The distinct linear functions that map [−1, 1] onto [0, 2] are
Let the function f: R → R be defined by f(x) = 4x – 1, ∀ x ∈ R. Then, show that f is one-one.
Set A has 3 elements and the set B has 4 elements. Then the number of injective mappings that can be defined from A to B is ______.
The domain of the function f: R → R defined by f(x) = `sqrt(x^2 - 3x + 2)` is ______
Let A be a finite set. Then, each injective function from A into itself is not surjective.
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
g = {(1, 4), (2, 4), (3, 4)}
Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:
h(x) = x|x|
Using the definition, prove that the function f: A→ B is invertible if and only if f is both one-one and onto
The function f: R → R defined as f(x) = x3 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 : N → R be defined by f(x) = x2. Range of the function among the following 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.
- The function f: Z → Z defined by f(x) = x2 is ____________.
Let f: R → R defined by f(x) = x4. Choose the correct answer
Let x is a real number such that are functions involved are well defined then the value of `lim_(t→0)[max{(sin^-1 x/3 + cos^-1 x/3)^2, min(x^2 + 4x + 7)}]((sin^-1t)/t)` where [.] is greatest integer function and all other brackets are usual brackets.
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)
If f : R `rightarrow` R is defined by `f(x) = (2x - 7)/4`, show that f(x) is one-one and onto.
The function defined by \[\mathrm{f}(x)=\frac{2x+3}{3x+4},x\neq-\frac{4}{3}\] is
