Advertisements
Advertisements
प्रश्न
Let f : N → N be defined by f(n) = `{((n+1)/2", if n is odd"),(n/2", if n is even"):}` for all n ∈ N.
State whether the function f is bijective. Justify your answer.
Advertisements
उत्तर
f : N → N is defined as f(n) = `{((n + 1)/2",", "if n is odd"), (n/2",", "if n is even"):}` for all n ∈ N.
It can be observed that:
f(1) = `(1+1)/2` = 1 and f(2) = `2/2` = 1 ...[By definition of f]
∴ f(1) = f(2), where 1 ≠ 2
∴ f is not one-one.
Consider a natural number (n) in co-domain N.
Case I: n is odd
∴ n = 2r + 1 for some r ∈ N. Then, there exists 4r + 1 ∈ N such that
f(4r + 1) = `(4r + 1 + 1)/2`
= 2r + 1
Case II: n is even
∴ n = 2r for some r ∈ N. Then, there exists 4r ∈ N such that
f(4r) = `(4r)/2`
= 2r
∴ f is onto.
Hence, f is not a bijective function.
APPEARS IN
संबंधित प्रश्न
Show that the function f : R* → R* defined by f(x) = `1/x` is one-one and onto, where R* is the set of all non-zero real numbers. Is the result true, if the domain R* is replaced by N with co-domain being same as R?
Check the injectivity and surjectivity of the following function:
f : N → N given by f(x) = x3
In the following case, state whether the function is one-one, onto or bijective. Justify your answer.
f : R → R defined by f(x) = 3 – 4x
Show that the function f : R → R given by f(x) = x3 is injective.
Given examples of two functions f: N → N and g: N → N such that gof is onto but f is not onto.
(Hint: Consider f(x) = x + 1 and `g(x) = {(x-1, ifx >1),(1, if x = 1):}`
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = 3 − 4x
Show that f : R→ R, given by f(x) = x — [x], is neither one-one nor onto.
Let f = {(3, 1), (9, 3), (12, 4)} and g = {(1, 3), (3, 3) (4, 9) (5, 9)}. Show that gof and fog are both defined. Also, find fog and gof.
Let f : R → R and g : R → R be defined by f(x) = x2 and g(x) = x + 1. Show that fog ≠ gof.
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).
State with reason whether the following functions have inverse :
f : {1, 2, 3, 4} → {10} with f = {(1, 10), (2, 10), (3, 10), (4, 10)}
Let A = {1, 2, 3, 4}; B = {3, 5, 7, 9}; C = {7, 23, 47, 79} and f : A → B, g : B → C be defined as f(x) = 2x + 1 and g(x) = x2 − 2. Express (gof)−1 and f−1 og−1 as the sets of ordered pairs and verify that (gof)−1 = f−1 og−1.
If f : R → R be defined by f(x) = x3 −3, then prove that f−1 exists and find a formula for f−1. Hence, find f−1(24) and f−1 (5).
Let A = R - {3} and B = R - {1}. Consider the function f : A → B defined by f(x) = `(x-2)/(x-3).`Show that f is one-one and onto and hence find f-1.
[CBSE 2012, 2014]
If f : R → R, g : R → are given by f(x) = (x + 1)2 and g(x) = x2 + 1, then write the value of fog (−3).
Let A = {x ∈ R : −4 ≤ x ≤ 4 and x ≠ 0} and f : A → R be defined by \[f\left( x \right) = \frac{\left| x \right|}{x}\]Write the range of f.
Write whether f : R → R, given by `f(x) = x + sqrtx^2` is one-one, many-one, onto or into.
If f : R → R is defined by f(x) = 3x + 2, find f (f (x)).
Let f : R → R be the function defined by f(x) = 4x − 3 for all x ∈ R Then write f . [NCERT EXEMPLAR]
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,
The function \[f : [0, \infty ) \to \text {R given by } f\left( x \right) = \frac{x}{x + 1} is\]
If \[f : R \to R is given by f\left( x \right) = 3x - 5, then f^{- 1} \left( x \right)\]
Let \[f\left( x \right) = \frac{\alpha x}{x + 1}, x \neq - 1\] Then, for what value of α is \[f \left( f\left( x \right) \right) = x?\]
A function f: R→ R defined by f(x) = `(3x) /5 + 2`, x ∈ R. Show that f is one-one and onto. Hence find f−1.
Let the function f: R → R be defined by f(x) = 4x – 1, ∀ x ∈ R. Then, show that f is one-one.
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 A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:
h(x) = x|x|
The function f : A → B defined by f(x) = 4x + 7, x ∈ R is ____________.
Let A = R – {3}, B = R – {1}. Let f : A → B be defined by `"f"("x") = ("x" - 2)/("x" - 3)` Then, ____________.
The mapping f : N → N is given by f(n) = 1 + n2, n ∈ N when N is the set of natural numbers is ____________.
Let f : R `->` R be a function defined by f(x) = x3 + 4, then f 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 : N → R be defined by f(x) = x2. Range of the function among the following is ____________.
If `f : R -> R^+ U {0}` be defined by `f(x) = x^2, x ∈ R`. The mapping is
A function f: x → y is said to be one – one (or injective) if:
'If 'f' is a linear function satisfying f[x + f(x)] = x + f(x), then f(5) can be equal to:
`x^(log_5x) > 5` implies ______.
The function f(x) = [x], where [x] denotes the greatest integer less than or equal to x; is continuous at ______.
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.
Which real-life situation resembles a one-one idea when no roll number is repeated?
