Advertisements
Advertisements
प्रश्न
Let f: R → R be the function defined by f(x) = 2x – 3 ∀ x ∈ R. write f–1
Advertisements
उत्तर
Given f(x) = 2x – 3 ∀ x ∈ R
Now, Leta, b ∈ R such that
f(a) = f(b)
⇒ 2a – 3 = 2b – 3
⇒ a = b
⇒ f(x) is one – one.
Also, If x, y ∈ R such that
f(x) = y
⇒ 2x – 3 = y
⇒ x = `(y + 3)/2` = (y) ∀ y ∈ R
⇒ f(x) is onto and therefore is bijective implies f(x) has an inverse
Let f–1 denote the inverse of f(x) then
f–1(x) = g(x)
= `(x + 3)/2` ∀ x ∈ R
APPEARS IN
संबंधित प्रश्न
In the following case, state whether the function is one-one, onto or bijective. Justify your answer.
f : R → R defined by f(x) = 1 + x2
Let A and B be sets. Show that f : A × B → B × A such that f(a, b) = (b, a) is a bijective function.
Let f : R → R be defined as f(x) = x4. Choose the correct answer.
Let f : R → R be defined as f(x) = 3x. Choose the correct answer.
Show that the function f : R → R given by f(x) = x3 is injective.
Give examples of two functions f: N → Z and g: Z → Z such that g o f is injective but gis not injective.
(Hint: Consider f(x) = x and g(x) =|x|)
Find the number of all onto functions from the set {1, 2, 3, ..., n} to itself.
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = sinx
Let A = [-1, 1]. Then, discuss whether the following function from A to itself is one-one, onto or bijective : g(x) = |x|
If A = {1, 2, 3}, show that a onto function f : A → A must be one-one.
Give examples of two surjective functions f1 and f2 from Z to Z such that f1 + f2 is not surjective.
Consider f : N → N, g : N → N and h : N → R defined as f(x) = 2x, g(y) = 3y + 4 and h(z) = sin z for all x, y, z ∈ N. Show that ho (gof) = (hog) of.
Find fog and gof if : f (x) = x+1, g (x) = sin x .
If f(x) = 2x + 5 and g(x) = x2 + 1 be two real functions, then describe each of the following functions:
(1) fog
(2) gof
(3) fof
(4) f2
Also, show that fof ≠ f2
Let f be any real function and let g be a function given by g(x) = 2x. Prove that gof = f + f.
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.
If f : C → C is defined by f(x) = (x − 2)3, write f−1 (−1).
Write the domain of the real function
`f (x) = 1/(sqrt([x] - x)`.
If f : R → R be defined by f(x) = (3 − x3)1/3, then find fof (x).
\[f : R \to R \text{given by} f\left( x \right) = x + \sqrt{x^2} \text{ is }\]
Mark the correct alternative in the following question:
Let f : R → R be given by f(x) = tanx. Then, f-1(1) is
Let A = R − (2) and B = R − (1). If f: A ⟶ B is a function defined by`"f(x)"=("x"-1)/("x"-2),` how that f is one-one and onto. Hence, find f−1.
If A = {a, b, c, d} and f = {a, b), (b, d), (c, a), (d, c)}, show that f is one-one from A onto A. Find f–1
Let R be the set of real numbers and f: R → R be the function defined by f(x) = 4x + 5. Show that f is invertible and find f–1.
Let N be the set of natural numbers and the function f: N → N be defined by f(n) = 2n + 3 ∀ n ∈ N. Then f is ______.
Let f: R → R be defined by f(x) = 3x – 4. Then f–1(x) is given by ______.
Let D be the domain of the real valued function f defined by f(x) = `sqrt(25 - x^2)`. Then, write D
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 f : R → R be defind by f(x) = `1/"x" AA "x" in "R".` Then f is ____________.
The mapping f : N → N is given by f(n) = 1 + n2, n ∈ N when N is the set of natural numbers 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 f: R → R be defined by f(x) = x2 is:
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 f: N → N be defined by f(x) = x2 is ____________.
The solution set of the inequation log1/3(x2 + x + 1) + 1 > 0 is ______.
Which one of the following graphs is a function of x?
 |
 |
| Graph A | Graph B |
If f : R `rightarrow` R is defined by `f(x) = (2x - 7)/4`, show that f(x) is one-one and onto.
The given function f : R → R is not ‘onto’ function. Give reason.
