Advertisements
Advertisements
प्रश्न
If f : R → R be the function defined by f(x) = 4x3 + 7, show that f is a bijection.
Advertisements
उत्तर
Injectivity:
Let x and y be any two elements in the domain (R), such that f(x) = f(y)
⇒ 4x3+7 = 4y3+ 7
⇒ 4x3= 4y3
⇒ x3= y3
⇒ x = y
So, f is one-one.
Surjectivity:
Let y be any element in the co-domain (R), such that f(x) = y for some element x in R (domain)
f(x) = y
⇒ 4x3+7 = y
⇒ 4x3= y −7
⇒ `x^3 = (y - 7)/4`
⇒ `x = 3sqrt(y-7)/4 in R`
So, for every element in the co-domain, there exists some pre-image in the domain. - f is onto.
Since, f is both one-to-one and onto, it is a bijection.
APPEARS IN
संबंधित प्रश्न
Show that the Signum Function f : R → R, given by `f(x) = {(1", if" x > 0), (0", if" x = 0), (-1", if" x < 0):}` is neither one-one nor onto.
Let A = R − {3} and B = R − {1}. Consider the function f : A → B defined by f(x) = `((x- 2)/(x -3))`. Is f one-one and onto? Justify your answer.
Prove that the function f : N → N, defined by f(x) = x2 + x + 1, is one-one but not onto
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = 5x3 + 4
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = `x/(x^2 +1)`
Let A = [-1, 1]. Then, discuss whether the following functions from A to itself is one-one, onto or bijective : h(x) = x2
If A = {1, 2, 3}, show that a one-one function f : A → A must be onto.
Give examples of two one-one functions f1 and f2 from R to R, such that f1 + f2 : R → R. defined by (f1 + f2) (x) = f1 (x) + f2 (x) is not one-one.
Find fog and gof if : f(x)= x + 1, g (x) = 2x + 3 .
If f(x) = sin x and g(x) = 2x be two real functions, then describe gof and fog. Are these equal functions?
Consider the function f : R+ → [-9 , ∞ ]given by f(x) = 5x2 + 6x - 9. Prove that f is invertible with f -1 (y) = `(sqrt(54 + 5y) -3)/5` [CBSE 2015]
Which one of the following graphs represents a function?
If f : C → C is defined by f(x) = (x − 2)3, write f−1 (−1).
If f : R → R defined by f(x) = 3x − 4 is invertible, then write f−1 (x).
Write the domain of the real function
`f (x) = sqrtx - [x] .`
Write the domain of the real function
`f (x) = sqrt([x] - x) .`
If f : {5, 6} → {2, 3} and g : {2, 3} → {5, 6} are given by f = {(5, 2), (6, 3)} and g = {(2, 5), (3, 6)}, then find fog. [NCERT EXEMPLAR]
Let f : R → R be the function defined by f(x) = 4x − 3 for all x ∈ R Then write f . [NCERT EXEMPLAR]
If f(x) = 4 −( x - 7)3 then write f-1 (x).
Let\[A = \left\{ x \in R : - 1 \leq x \leq 1 \right\} = \text{B and C} = \left\{ x \in R : x \geq 0 \right\} and\]\[S = \left\{ \left( x, y \right) \in A \times B : x^2 + y^2 = 1 \right\} \text{and } S_0 = \left\{ \left( x, y \right) \in A \times C : x^2 + y^2 = 1 \right\}\]
Then,
\[f : Z \to Z\] be given by
` f (x) = {(x/2, ", if x is even" ) ,(0 , ", if x is odd "):}`
Then, f is
Let \[f\left( x \right) = x^2 and g\left( x \right) = 2^x\] Then, the solution set of the equation
If \[f : R \to R is given by f\left( x \right) = 3x - 5, then f^{- 1} \left( x \right)\]
If the function
\[f : R \to R\] be such that
\[f\left( x \right) = x - \left[ x \right]\] where [x] denotes the greatest integer less than or equal to x, then \[f^{- 1} \left( x \right)\]
The domain of the function f: R → R defined by f(x) = `sqrt(x^2 - 3x + 2)` is ______
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
k = {(1,4), (2, 5)}
Using the definition, prove that the function f: A→ B is invertible if and only if f is both one-one and onto
Let g(x) = x2 – 4x – 5, then ____________.
The function f : R → R given by f(x) = x3 – 1 is ____________.
Let f : [0, ∞) → [0, 2] be defined by `"f" ("x") = (2"x")/(1 + "x"),` then f is ____________.
If N be the set of all-natural numbers, consider f: N → N such that f(x) = 2x, ∀ x ∈ N, then f is ____________.
Students of Grade 9, planned to plant saplings along straight lines, parallel to each other to one side of the playground ensuring that they had enough play area. Let us assume that they planted one of the rows of the saplings along the line y = x − 4. Let L be the set of all lines which are parallel on the ground and R be a relation on L.
Answer the following using the above information.
- The function f: R → R defined by f(x) = x − 4 is ____________.
A function f: x → y is said to be one – one (or injective) if:
Let f: R→R be defined as f(x) = 2x – 1 and g: R – {1}→R be defined as g(x) = `(x - 1/2)/(x - 1)`. Then the composition function f (g(x)) is ______.
Let f(x) be a polynomial of degree 3 such that f(k) = `-2/k` for k = 2, 3, 4, 5. Then the value of 52 – 10f(10) is equal to ______.
Let A = R – {2} and B = R – {1}. If f: A `→` B is a function defined by f(x) = `(x - 1)/(x - 2)` then show that f is a one-one and an onto function.
The function defined by \[\mathrm{f}(x)=\frac{2x+3}{3x+4},x\neq-\frac{4}{3}\] is
