Advertisements
Advertisements
प्रश्न
Let f: A → B and g: B → C be the bijective functions. Then (g o f)–1 is ______.
विकल्प
f –1 o g–1
f o g
g–1 o f–1
g o f
Advertisements
उत्तर
Let f: A → B and g: B → C be the bijective functions. Then (g o f)–1 is f –1 o g–1.
Explanation:
Given that, f: A → B and g: B → C be the bijective functions.
(f –1 o g–1) o (g o f) = f –1 o (g–1 o g o f)
= f –1 o (g–1 o g) o f ......(As composition of functions is associative)
= f –1 o IB o f) .......(Where IB is identity function on B)
= (f –1 o IB) o f
= f –1 o f
= IA
Thus (g o f)–1 = f –1 o g –1
APPEARS IN
संबंधित प्रश्न
If the function f : R → R be defined by f(x) = 2x − 3 and g : R → R by g(x) = x3 + 5, then find the value of (fog)−1 (x).
Let f : W → W be defined as
`f(n)={(n-1, " if n is odd"),(n+1, "if n is even") :}`
Show that f is invertible a nd find the inverse of f. Here, W is the set of all whole
numbers.
Let f: {1, 3, 4} → {1, 2, 5} and g: {1, 2, 5} → {1, 3} be given by f = {(1, 2), (3, 5), (4, 1)} and g = {(1, 3), (2, 3), (5, 1)}. Write down gof.
Let f, g and h be functions from R to R. Show that
(f + g)oh = foh + goh
(f · g)oh = (foh)·(goh)
State with reason whether following functions have inverse
h: {2, 3, 4, 5} → {7, 9, 11, 13} with h = {(2, 7), (3, 9), (4, 11), (5, 13)}
Let f: X → Y be an invertible function. Show that f has unique inverse. (Hint: suppose g1 and g2 are two inverses of f. Then for all y ∈ Y, fog1(y) = IY(y) = fog2(y). Use one-one ness of f).
Let f: X → Y be an invertible function. Show that the inverse of f−1 is f, i.e., (f−1)−1 = f.
Let `f: R - {-4/3} → R` be a function defined as `f(x) = (4x)/(3x + 4)`. The inverse of f is map g: Range `f → R - {-4/3}` given by
If f: R → R is defined by f(x) = x2 − 3x + 2, find f(f(x)).
Consider f: `R_+ -> [-5, oo]` given by `f(x) = 9x^2 + 6x - 5`. Show that f is invertible with `f^(-1) (y) ((sqrt(y + 6)-1)/3)`
Hence Find
1) `f^(-1)(10)`
2) y if `f^(-1) (y) = 4/3`
where R+ is the set of all non-negative real numbers.
Let f : W → W be defined as f(x) = x − 1 if x is odd and f(x) = x + 1 if x is even. Show that f is invertible. Find the inverse of f, where W is the set of all whole numbers.
Let f: R → R be defined by f(x) = 3x 2 – 5 and g: R → R by g(x) = `x/(x^2 + 1)` Then gof is ______.
Let f: R → R be the function defined by f(x) = sin (3x+2) ∀ x ∈ R. Then f is invertible.
The composition of functions is associative.
If f(x) = (ax2 + b)3, then the function g such that f(g(x)) = g(f(x)) is given by ____________.
Let f : R – `{3/5}`→ R be defined by f(x) = `(3"x" + 2)/(5"x" - 3)` Then ____________.
If f(x) = (ax2 – b)3, then the function g such that f{g(x)} = g{f(x)} is given by ____________.
Which one of the following functions is not invertible?
If f is an invertible function defined as f(x) `= (3"x" - 4)/5,` then f-1(x) 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}
- Two neighbors X and Y ∈ I. X exercised his voting right while Y did not cast her vote in a general election - 2019. Which of the following is true?
`f : x -> sqrt((3x^2 - 1)` and `g : x -> sin (x)` then `gof : x ->`?
The domain of definition of f(x) = log x2 – x + 1) (2x2 – 7x + 9) is:-
Domain of the function defined by `f(x) = 1/sqrt(sin^2 - x) log_10 (cos^-1 x)` is:-
Let A = `{3/5}` and B = `{7/5}` Let f: A → B: f(x) = `(7x + 4)/(5x - 3)` and g:B → A: g(y) = `(3y + 4)/(5y - 7)` then (gof) is equal to
If f: A → B and G B → C are one – one, then g of A → C is
If f: N → Y be a function defined as f(x) = 4x + 3, Where Y = {y ∈ N: y = 4x+ 3 for some x ∈ N} then function is
Let `f : R {(-1)/3} → R - {0}` be defined as `f(x) = 5/(3x + 1)` is invertible. Find f–1(x).
Which statement correctly describes a one-one (injective) function?
Which statement correctly describes an onto (surjective) function?
Which equation expresses that applying a function and then its inverse returns the original input?
According to the Reflective Property, the graph of \(f^{-1}\) is the exact reflection of the graph of \(f\) across which line?
Which pair consists of examples of self-inverse functions?
For \(f(x)=4x+3\), where \(Y=\{y\in\mathbb{N}:y=4x+3\text{ for some }x\in\mathbb{N}\}\), what is the inverse function \(g:Y\to\mathbb{N}\)?
For \(f:\mathbb{R}\to\mathbb{R}\) defined by \(f(x)=10x+7\), which function satisfies \(g\circ f=f\circ g=I_R\)?
For \(f:\mathbb{R}\to(-1,1)\) defined by \(f(x)=\frac{e^x-e^{-x}}{e^x+e^{-x}}\), what is \(f^{-1}(x)\)?
In the graphical example, what are the domain and range of the given function?
