Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
f: `R_+ -> [-5, oo]` given by `f(x) = 9x^2 + 6x - 5`
To show: f is one-one and onto.
Let us assume that f is not one-one.
Therefore there exist two or more numbers for which images are same.
For x1, x2 ∈ R+ and x1 ≠ x2
Since x1 and x2 are positive,
9(x1 + x2) + 6 > 0
∴ x1 − x2 = 0⇒ x1 = x2
Therefore, it contradicts our assumption.
Hence the function f is one-one.
Now, let is prove that f is onto.
A function f : X → Y is onto if for every y ∈ Y, there exist a pre-image in X.
f(x) = 9x2 + 6x −5
= 9x2 + 6x + 1 - 6
=(3x + 1)2 - 6
Now, for all x ∈ R+ or [0,∞), f(x) ∈ [−5, ∞)
∴ Range = co-domain.
Hence, f is onto.
Therefore, function f is invertible.
Now, let y = 9x2 + 6x − 5
1) `f^(-1) (10) = (sqrt(10+6)-1)/3 = (4-1)/3 = 1`
2) `f^(-1) (y) = 4/3`
`:.(sqrt(y + 6) -1)/3 = 4/3`
`=> sqrt(y + 6)-1 = 4`
`=> sqrt(y+6) = 5`
`=> y + 6 = 25`
=> y = 19
APPEARS IN
संबंधित प्रश्न
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?
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 → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f.
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.
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: 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 = ______.
The composition of functions is commutative.
The composition of functions is associative.
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 ____________.
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.
If f : R → R defined by f(x) `= (3"x" + 5)/2` is an invertible function, then find f-1.
Domain of the function defined by `f(x) = 1/sqrt(sin^2 - x) log_10 (cos^-1 x)` 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).
A function is called a self-inverse function when which condition holds?
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?
What are the domain and range of the inverse function in the graphical example?
