Advertisements
Advertisements
प्रश्न
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
विकल्प
`g(y) = (3y)/(3 - 4y)`
`g(y) = (4y)/(4 - 3y)`
`g(y) = (4y)/(3 - 4y)`
`g(y) = (3y)/(4 - 3y)`
Advertisements
उत्तर
`bb(g(y) = (4y)/(4 - 3y))`
Explanation:
It is given that `f: R -{-4/3} → R` is defined as `f(x) = (4x)/(3x - 4)`.
Let y be an arbitrary element of Range f.
Then, there exists `x ∈ R - {-4/3}` such that y = f(x).
⇒ `y = (4x)/(3x + 4)`
⇒ 3xy + 4y = 4x
⇒ x(4 – 3y) = 4y
⇒ `x = (4y)/(4 - 3y)`
Let us define g: Range `f → R -{-4/3}` as `g(y) = (4y)/(4 - 3y)`.
Now, (gof)(x) = g(f(x))
= `g((4x)/(3x + 4))`
= `(4((4x)/(3x + 4)))/(4 - 3((4x)/(3x + 4)))`
= `(16x)/(12x + 16 - 12x)`
= `(16x)/16`
= x
And (fog)(y) = f(g(y))
= `f((4y)/(4 - 3y))`
= `(4((4y)/(4 - 3y)))/(3((4y)/(4 - 3y)) + 4)`
= `(16y)/(12y + 16 - 12y)`
= `(16y)/16`
= y
∴ `gof = I_(R - {-4/3})` and `fog = I_"Range f"`
Thus, g is the inverse of f i.e., f−1 = g.
Hence, the inverse of f is the map g: Range `f → R - {-4/3}`, which is given by `g(y) = (4y)/(4 - 3y)`.
संबंधित प्रश्न
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, g and h be functions from R to R. Show that
(f + g)oh = foh + goh
(f · g)oh = (foh)·(goh)
If `f(x) = (4x + 3)/(6x - 4), x ≠ 2/3` show that fof(x) = x, for all `x ≠ 2/3`. What is the inverse of f?
State with reason whether following functions have inverse
g: {5, 6, 7, 8} → {1, 2, 3, 4} with g = {(5, 4), (6, 3), (7, 4), (8, 2)}
Show that f: [–1, 1] → R, given by f(x) = `x/(x + 2)` is one-one. Find the inverse of the function f: [–1, 1] → Range f.
(Hint: For y in Range f, y = `f(x) = x/(x + 2)` for some x in [–1, 1] i.e., `x = (2y)/(1 - y)`)
Consider f: R+ → [4, ∞) given by f(x) = x2 + 4. Show that f is invertible with the inverse f−1 of given f by `f^(-1)(y) = sqrt(y - 4)`, where R+ is the set of all non-negative real numbers.
Consider f: R+ → [–5, ∞) given by f(x) = 9x2 + 6x – 5. Show that f is invertible with `f^(-1)(y) = ((sqrt(y + 6) - 1)/3)`.
Let f: X → Y be an invertible function. Show that the inverse of f−1 is f, i.e., (f−1)−1 = f.
If f: R → R be given by `f(x) = (3 - x^3)^(1/3)`, then fof(x) is ______.
Is g = {(1, 1), (2, 3), (3, 5), (4, 7)} a function? If g is described by g (x) = αx + β, then what value should be assigned to α and β
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: [0, 1] → [0, 1] be defined by f(x) = `{{:(x",", "if" x "is rational"),(1 - x",", "if" x "is irrational"):}`. Then (f o f) x is ______.
If f(x) = (ax2 + b)3, then the function g such that f(g(x)) = g(f(x)) is given by ____________.
Let f : N → R : f(x) = `((2"x"−1))/2` and g : Q → R : g(x) = x + 2 be two functions. Then, (gof) `(3/2)` is ____________.
Let f : R → R be the functions defined by f(x) = x3 + 5. Then f-1(x) is ____________.
Let f : R – `{3/5}`→ R be defined by f(x) = `(3"x" + 2)/(5"x" - 3)` Then ____________.
If f : R → R defind by f(x) = `(2"x" - 7)/4` is an invertible function, then find f-1.
If f : R → R defined by f(x) `= (3"x" + 5)/2` is an invertible function, then find f-1.
`f : x -> sqrt((3x^2 - 1)` and `g : x -> sin (x)` then `gof : x ->`?
If f: A → B and G B → C are one – one, then g of A → C is
Let `f : R {(-1)/3} → R - {0}` be defined as `f(x) = 5/(3x + 1)` is invertible. Find f–1(x).
A function \(f:X\to Y\) is defined to be invertible if there exists a function \(g:Y\to X\) such that which conditions hold?
Which statement correctly describes a one-one (injective) function?
Which statement correctly describes an onto (surjective) function?
A function that is both one-one and onto is called what?
What is the necessary and sufficient condition for a function to be invertible?
Which property of inverse functions is represented by \((f^{-1})^{-1}=f\)?
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 \(f:\mathbb{R}\to\mathbb{R}\) defined by \(f(x)=10x+7\), which function satisfies \(g\circ f=f\circ g=I_R\)?
In the graphical example, what are the domain and range of the given function?
Which statement expresses that an inverse is unique whenever it exists?
