Advertisements
Advertisements
प्रश्न
Find fog and gof if : f(x) = `x^2` + 2 , g (x) = 1 − `1/ (1-x)`.
Advertisements
उत्तर
f (x) = x2+ 2
f : R → [ 2, ∞ )
g (x) = 1 `- 1/(1-x)`
For domain of g : 1− x ≠ 0
⇒ x ≠ 1
⇒ Domain of g = R−{1}
g (x )= `1 - 1/(1-x) = (1-x-1)/(1-x) = (-x)/(1-x)`
For range of g :
`y = (- x)/ (1-x)`
⇒ y − xy = − x
⇒ y = xy − x
⇒ y = x (y−1)
⇒ `x = y/(y-1)`
Range of g =R−{1}
So, g : R−{1}→R−{1}
Computing fog :
Clearly, the range of g is a subset of the domain of f.
⇒ fog : R − {1}→ R
(fog) (x) = f (g (x))
`= f ((-x)/ (x-1) )`
`= ((-x)/ (x-1))^2 + 2`
`=(x^2 + 2x^2 +2-4x)/(1-x)^2`
`= (3x^2-4x +2 )/ (1-x)^2`
Computing gof :
Clearly, the range of f is a subset of the domain of g.
⇒ gof : R→R
(gof) (x) = g (f (x))
= g ( x2 + 2 )
`= 1- 1/(1-(x^2 + 2))`
`= - 1/(1-(x^2 + 2))`
`= (x^2 + 2)/(x^2 + 1)`
APPEARS IN
संबंधित प्रश्न
Check the injectivity and surjectivity of the following function:
f : R → R given by f(x) = x2
Check the injectivity and surjectivity of the following function:
f : N → N given by f(x) = x3
Show that the function f : R → R given by f(x) = x3 is injective.
Give an example of a function which is one-one but not onto ?
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = 3 − 4x
If A = {1, 2, 3}, show that a onto function f : A → A must be one-one.
If f : A → B and g : B → C are one-one functions, show that gof is a one-one function.
Find fog and gof if : f (x) = x+1, g (x) = sin x .
If f : R → (−1, 1) defined by `f (x) = (10^x- 10^-x)/(10^x + 10 ^-x)` is invertible, find f−1.
Which one of the following graphs represents a function?
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 → R+ be defined by f(x) = ax, a > 0 and a ≠ 1. Write f−1 (x).
Let f : R → R be defined as `f (x) = (2x - 3)/4.` write fo f-1 (1) .
If f(x) = x + 7 and g(x) = x − 7, x ∈ R, write fog (7).
What is the range of the function
`f (x) = ([x - 1])/(x -1) ?`
\[f : A \to \text{B given by } 3^{ f\left( x \right)} + 2^{- x} = 4\] is a bijection, then
The function f : R → R defined by
`f (x) = 2^x + 2^(|x|)` is
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
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
If a function\[f : [2, \infty )\text{ to B defined by f}\left( x \right) = x^2 - 4x + 5\] is a bijection, then B =
The function f : [-1/2, 1/2, 1/2] → [-π /2,π/2], defined by f (x) = `sin^-1` (3x - `4x^3`), is
If \[g \left( f \left( x \right) \right) = \left| \sin x \right| \text{and} f \left( g \left( x \right) \right) = \left( \sin \sqrt{x} \right)^2 , \text{then}\]
Let
\[A = \left\{ x \in R : x \geq 1 \right\}\] The inverse of the function,
\[f : A \to A\] given by
\[f\left( x \right) = 2^{x \left( x - 1 \right)} , is\]
Mark the correct alternative in the following question:
If the set A contains 7 elements and the set B contains 10 elements, then the number one-one functions from A to B is
Let A = R – {3}, B = R – {1}. Let f: A → B be defined by f(x) = `(x - 2)/(x - 3)` ∀ x ∈ A . Then show that f is bijective.
Let f: R → R be the functions defined by f(x) = x3 + 5. Then f–1(x) is ______.
Let f : R → R be a function defined by f(x) `= ("e"^abs"x" - "e"^-"x")/("e"^"x" + "e"^-"x")` then f(x) is
An organization conducted a bike race under 2 different categories-boys and girls. Totally there were 250 participants. Among all of them finally, three from Category 1 and two from Category 2 were selected for the final race. Ravi forms two sets B and G with these participants for his college project. Let B = {b1,b2,b3} G={g1,g2} where B represents the set of boys selected and G the set of girls who were selected for the final race.
Ravi decides to explore these sets for various types of relations and functions.
- Let R: B → G be defined by R = { (b1,g1), (b2,g2),(b3,g1)}, then R is ____________.
If f: R → R given by f(x) =(3 − x3)1/3, find f0f(x)
'If 'f' is a linear function satisfying f[x + f(x)] = x + f(x), then f(5) can be equal to:
If log102 = 0.3010.log103 = 0.4771 then the number of ciphers after decimal before a significant figure comes in `(5/3)^-100` is ______.
If f: [0, 1]→[0, 1] is defined by f(x) = `(x + 1)/4` and `d/(dx) underbrace(((fofof......of)(x)))_("n" "times")""|_(x = 1/2) = 1/"m"^"n"`, m ∈ N, then the value of 'm' is ______.
The domain of function is f(x) = `sqrt(-log_0.3(x - 1))/sqrt(x^2 + 2x + 8)` is ______.
Let f(x) = ax (a > 0) be written as f(x) = f1(x) + f2(x), where f1(x) is an even function and f2(x) is an odd function. Then f1(x + y) + f1(x – y) equals ______.
The graph of the function y = f(x) is symmetrical about the line x = 2, then ______.
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.
