Advertisements
Advertisements
Question
Which of the following functions from Z into Z are bijections?
Options
f(x) = x3
f(x) = x + 2
f(x) = 2x + 1
f(x) = x2 + 1
Advertisements
Solution
f(x) = x + 2
Explanation:
For bijection on Z, f(x) must be one-one and onto
Function f(x) = x2 + 1 is many-one as f(1) = f(–1)
Range of f(x) = x3 is not Z for x ∈ Z.
Also f(x) = 2x + 1 takes only values of types = 2k + 1 for x = k ∈ Z
But f(x) = x + 2 takes all integral values for x ∈ Z
Hence f(x)= x + 2 is bijection on Z
APPEARS IN
RELATED QUESTIONS
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 and B be sets. Show that f : A × B → B × A such that f(a, b) = (b, a) is bijective function.
Let f : R → R be defined as f(x) = x4. Choose the correct 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) = |x|
Classify the following function as injection, surjection or bijection :
f : Q − {3} → Q, defined by `f (x) = (2x +3)/(x-3)`
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.
Give examples of two surjective functions f1 and f2 from Z to Z such that f1 + f2 is not surjective.
Find gof and fog when f : R → R and g : R → R is defined by f(x) = x and g(x) = |x| .
Find gof and fog when f : R → R and g : R → R is defined by f(x) = 8x3 and g(x) = x1/3.
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.
Verify associativity for the following three mappings : f : N → Z0 (the set of non-zero integers), g : Z0 → Q and h : Q → R given by f(x) = 2x, g(x) = 1/x and h(x) = ex.
Find fog and gof if : f (x) = x2 g(x) = cos x .
Let f be any real function and let g be a function given by g(x) = 2x. Prove that gof = f + f.
if f (x) = `sqrt (x +3) and g (x) = x ^2 + 1` be two real functions, then find fog and gof.
If f : R → R is defined by f(x) = x2, write f−1 (25)
If f : R → R is defined by f(x) = 3x + 2, find f (f (x)).
If a function g = {(1, 1), (2, 3), (3, 5), (4, 7)} is described by g(x) = \[\alpha x + \beta\] then find the values of \[\alpha\] and \[ \beta\] . [NCERT EXEMPLAR]
The function f : R → R defined by
`f (x) = 2^x + 2^(|x|)` is
If \[g \left( f \left( x \right) \right) = \left| \sin x \right| \text{and} f \left( g \left( x \right) \right) = \left( \sin \sqrt{x} \right)^2 , \text{then}\]
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 \[f^{- 1} \left( x \right)\]
If \[f\left( x \right) = \sin^2 x\] and the composite function \[g\left( f\left( x \right) \right) = \left| \sin x \right|\] then g(x) is equal to
If \[f : R \to R\] is given by \[f\left( x \right) = x^3 + 3, \text{then} f^{- 1} \left( x \right)\] is equal to
If f(x) = `(x+3)/(4x−5) , "g"(x) = (3+5x)/(4x−1)` then verify that `("fog") (x)` = x.
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 A = {1, 2, 3, ...n} and B = {a, b}. Then the number of surjections from A into B is ______.
Let f: R → R be the functions defined by f(x) = x3 + 5. Then f–1(x) is ______.
The smallest integer function f(x) = [x] is ____________.
Which of the following functions from Z into Z is bijective?
The function f : R → R given by f(x) = x3 – 1 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 ____________.
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:
The domain of the function `cos^-1((2sin^-1(1/(4x^2-1)))/π)` is ______.
The domain of function is f(x) = `sqrt(-log_0.3(x - 1))/sqrt(x^2 + 2x + 8)` is ______.
Let f(n) = `[1/3 + (3n)/100]n`, where [n] denotes the greatest integer less than or equal to n. Then `sum_(n = 1)^56f(n)` is equal to ______.
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 ______.
The function f(x) = [x], where [x] denotes the greatest integer less than or equal to x; is continuous at ______.
The function \(f:\mathbb{N}\to\mathbb{N}\) defined by \(f(x)=5x\) is:
Let \(X=\{-1,2,5,8,11\}\), \(Y=\{4,6,8\}\), and suppose different elements \(2,5,8\) of \(X\) have the same image \(4\). What type of function is it with respect to the one-one property?
