Advertisements
Advertisements
प्रश्न
Let f : R − {−1} → R − {1} be given by\[f\left( x \right) = \frac{x}{x + 1} . \text{Write } f^{- 1} \left( x \right)\]
Advertisements
उत्तर
\[Let f^{- 1} \left( x \right) = y . . . \left( 1 \right)\]
\[ \Rightarrow f\left( y \right) = x\]
\[ \Rightarrow \frac{y}{y + 1} = x\]
\[ \Rightarrow y = xy + x\]
\[ \Rightarrow y - xy = x\]
\[ \Rightarrow y\left( 1 - x \right) = x\]
\[ \Rightarrow y = \frac{x}{1 - x}\]
\[ \Rightarrow f^{- 1} \left( x \right) = \frac{x}{1 - x} [\text{from}\left( 1 \right)]\]
APPEARS IN
संबंधित प्रश्न
Check the injectivity and surjectivity of the following function:
f : N → N given by f(x) = x2
Check the injectivity and surjectivity of the following function:
f : R → R given by f(x) = x2
Give an example of a function which is not one-one but onto ?
Which of the following functions from A to B are one-one and onto ?
f3 = {(a, x), (b, x), (c, z), (d, z)} ; A = {a, b, c, d,}, B = {x, y, z}.
Show that the function f : R − {3} → R − {2} given by f(x) = `(x-2)/(x-3)` is a bijection.
If f : R → R be the function defined by f(x) = 4x3 + 7, show that f is a bijection.
If A = {1, 2, 3}, show that a onto function f : A → A must be one-one.
Find fog and gof if : f (x) = x+1, g (x) = sin x .
If f(x) = |x|, prove that fof = f.
Let f, g, h be real functions given by f(x) = sin x, g (x) = 2x and h (x) = cos x. Prove that fog = go (fh).
Let f be a real function given by f (x)=`sqrt (x-2)`
Find each of the following:
(i) fof
(ii) fofof
(iii) (fofof) (38)
(iv) f2
Also, show that fof ≠ `f^2` .
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 (x) = (sqrt (x +6)-1)/3 .`
If f : Q → Q, g : Q → Q are two functions defined by f(x) = 2 x and g(x) = x + 2, show that f and g are bijective maps. Verify that (gof)−1 = f−1 og −1.
Let f : [−1, ∞) → [−1, ∞) be given by f(x) = (x + 1)2 − 1, x ≥ −1. Show that f is invertible. Also, find the set S = {x : f(x) = f−1 (x)}.
Let f : R → R be defined as `f (x) = (2x - 3)/4.` write fo f-1 (1) .
Let
\[A = \left\{ x \in R : - 1 \leq x \leq 1 \right\} = B\] Then, the mapping\[f : A \to \text{B given by} f\left( x \right) = x\left| x \right|\] is
Which function is used to check whether a character is alphanumeric or not?
Let the function f: R → R be defined by f(x) = 4x – 1, ∀ x ∈ R. Then, show that f is one-one.
Show that the function f: R → R defined by f(x) = `x/(x^2 + 1)`, ∀ ∈ + R , is neither one-one nor onto
Let f: R → R be defined by f(x) = 3x – 4. Then f–1(x) is given by ______.
For sets A, B and C, let f: A → B, g: B → C be functions such that g o f is injective. Then both f and g are injective functions.
Let f: R → R be the function defined by f(x) = 2x – 3 ∀ x ∈ R. write f–1
Let X = {1, 2, 3}and Y = {4, 5}. Find whether the following subset of X ×Y are function from X to Y or not
g = {(1, 4), (2, 4), (3, 4)}
Let g(x) = x2 – 4x – 5, then ____________.
Let f : [0, ∞) → [0, 2] be defined by `"f" ("x") = (2"x")/(1 + "x"),` then f is ____________.
Range of `"f"("x") = sqrt((1 - "cos x") sqrt ((1 - "cos x")sqrt ((1 - "cos x")....infty))`
Raji visited the Exhibition along with her family. The Exhibition had a huge swing, which attracted many children. Raji found that the swing traced the path of a Parabola as given by y = x2.
Answer the following questions using the above information.
- Let : N → R be defined by f(x) = x2. Range of the function among the following is ____________.
A function f: x → y is/are called onto (or surjective) if x under f.
Function f: R → R, defined by f(x) = `x/(x^2 + 1)` ∀ x ∈ R is not
Number of integral values of x satisfying the inequality `(3/4)^(6x + 10 - x^2) < 27/64` is ______.
`x^(log_5x) > 5` implies ______.
Consider a set containing function A= {cos–1cosx, sin(sin–1x), sinx((sinx)2 – 1), etan{x}, `e^(|cosx| + |sinx|)`, sin(tan(cosx)), sin(tanx)}. B, C, D, are subsets of A, such that B contains periodic functions, C contains even functions, D contains odd functions then the value of n(B ∩ C) + n(B ∩ D) is ______ where {.} denotes the fractional part of functions)
The graph of the function y = f(x) is symmetrical about the line x = 2, then ______.
The trigonometric equation tan–1x = 3tan–1 a has solution for ______.
Let f: R→Rbe defined as f (x) = `(x^2 + 1)/2`, then ______.
A function is called many-one if two or more different elements of the domain have:
