Advertisements
Advertisements
Question
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
Options
`g(y) = (3y)/(3 - 4y)`
`g(y) = (4y)/(4 - 3y)`
`g(y) = (4y)/(3 - 4y)`
`g(y) = (3y)/(4 - 3y)`
Advertisements
Solution
`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)`.
RELATED QUESTIONS
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.
Find gof and fog, if f(x) = |x| and g(x) = |5x – 2|.
Find gof and fog, if f(x) = 8x3 and `g(x) = x^(1/3)`.
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?
Consider f: R → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f.
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)`.
If f: R → R be given by `f(x) = (3 - x^3)^(1/3)`, then fof(x) is ______.
Let f: W → W be defined as f(n) = n − 1, if is odd and f(n) = n + 1, if n is even. Show that f is invertible. Find the inverse of f. Here, W is the set of all whole 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.
If f : R → R, f(x) = x3 and g: R → R , g(x) = 2x2 + 1, and R is the set of real numbers, then find fog(x) and gof (x)
Let f: A → B and g: B → C be the bijective functions. Then (g o f)–1 is ______.
Let f: N → R be the function defined by f(x) = `(2x - 1)/2` and g: Q → R be another function defined by g(x) = x + 2. Then (g o f) `3/2` is ______.
Let f = {(1, 2), (3, 5), (4, 1) and g = {(2, 3), (5, 1), (1, 3)}. Then g o f = ______ and f o g = ______.
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 commutative.
The composition of functions is associative.
Every function is invertible.
If f : R → R, g : R → R and h : R → R is such that f(x) = x2, g(x) = tanx and h(x) = logx, then the value of [ho(gof)](x), if x = `sqrtpi/2` will be ____________.
If f(x) = `(3"x" + 2)/(5"x" - 3)` then (fof)(x) 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 ____________.
The inverse of the function `"y" = (10^"x" - 10^-"x")/(10^"x" + 10^-"x")` is ____________.
Consider the function f in `"A = R" - {2/3}` defiend as `"f"("x") = (4"x" + 3)/(6"x" - 4)` Find f-1.
`f : x -> sqrt((3x^2 - 1)` and `g : x -> sin (x)` then `gof : x ->`?
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: 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
If f(x) = [4 – (x – 7)3]1/5 is a real invertible function, then find f–1(x).
