Advertisements
Advertisements
प्रश्न
If f(x) = [4 – (x – 7)3]1/5 is a real invertible function, then find f–1(x).
Advertisements
उत्तर
Given f(x) = [4 – (x – 7)3]1/5 is a real invertible function.
Let f(x) = y
`\implies` y = [4 – (x – 7)3]1/5
`\implies` y5 = 4 – (x – 7)3
`\implies` (x – 7)3 = 4 – y5
`\implies` x – 7 = [4 – y5]1/3
`\implies` x = 7 + (4 – y5)1/3
`\implies` f1(y) = 7 + (4 – y5)1/3 ` {{:(∵ f(x) "is invertible"), (∴ f(x) = "y"),(\implies x = f^-1("y")):}`
Hence f1(x) = 7 + (4 – x5)1/3
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).
Find gof and fog, if f(x) = |x| and g(x) = |5x – 2|.
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+ → [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.
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).
Consider f: {1, 2, 3} → {a, b, c} given by f(1) = a, f(2) = b and f(3) = c. Find f−1 and show that (f−1)−1 = f.
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.
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.
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)
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: A → B and g: B → C be the bijective functions. Then (g o f)–1 is ______.
The composition of functions is commutative.
If f : R → R, g : R → R and h : R → R are such that f(x) = x2, g(x) = tan x and h(x) = log x, then the value of (go(foh)) (x), if x = 1 will be ____________.
If f(x) = `(3"x" + 2)/(5"x" - 3)` then (fof)(x) is ____________.
If f(x) = (ax2 – b)3, then the function g such that f{g(x)} = g{f(x)} is given by ____________.
Consider the function f in `"A = R" - {2/3}` defiend as `"f"("x") = (4"x" + 3)/(6"x" - 4)` Find f-1.
If f is an invertible function defined as f(x) `= (3"x" - 4)/5,` then f-1(x) 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
Let 'D' be the domain of the real value function on Ir defined by f(x) = `sqrt(25 - x^2)` the D is :-
