Advertisements
Advertisements
Question
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = x3 − x
Advertisements
Solution
f : R → R, defined by f(x) = x3 − x
Injection test :
Let x and y be any two elements in the domain (R), such that f(x) = f(y).
f(x) = f(y)
x3−x=y3−y
Here, we cannot say x=y.
For example, x=1 and y=-1
x3−x =1−1= 0
y3−y=(−1)3−(−1)−1+1=0
So, 1 and -1 have the same image 0 .
So, f is not an injection.
Surjection test :
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
x3 − x = y
By observation we can say that there exist some x in R, such that x3 - x = y.
So, f is a surjection and f is not a bijection.
APPEARS IN
RELATED QUESTIONS
Check the injectivity and surjectivity of the following function:
f : N → N given by f(x) = x3
Show that the function f : R → R given by f(x) = x3 is injective.
Classify the following function as injection, surjection or bijection : f : Z → Z given by f(x) = x3
If f : A → B is an injection, such that range of f = {a}, determine the number of elements in A.
Set of ordered pair of a function ? If so, examine whether the mapping is injective or surjective :{(a, b) : a is a person, b is an ancestor of a}
Find fog and gof if : f(x) = sin−1 x, g(x) = x2
Find fog and gof if : f (x) = x+1, g (x) = sin x .
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).
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.
Which of the following graphs represents a one-one function?
If f : C → C is defined by f(x) = x2, write f−1 (−4). Here, C denotes the set of all complex numbers.
Let f be a function from C (set of all complex numbers) to itself given by f(x) = x3. 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] .`
What is the range of the function
`f (x) = ([x - 1])/(x -1) ?`
The function
\[f : R \to R\] defined by\[f\left( x \right) = \left( x - 1 \right) \left( x - 2 \right) \left( x - 3 \right)\]
(a) one-one but not onto
(b) onto but not one-one
(c) both one and onto
(d) neither one-one nor onto
\[f : R \to R\] is defined by
\[f\left( x \right) = \frac{e^{x^2} - e^{- x^2}}{e^{x^2 + e^{- x^2}}} is\]
Which of the following functions from
\[A = \left\{ x \in R : - 1 \leq x \leq 1 \right\}\]
Let \[f\left( x \right) = \frac{1}{1 - x} . \text{Then}, \left\{ f o \left( fof \right) \right\} \left( x \right)\]
The distinct linear functions that map [−1, 1] onto [0, 2] are
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 f: R → R be the function defined by f(x) = 2x – 3 ∀ x ∈ R. write f–1
Are the following set of ordered pairs functions? If so, examine whether the mapping is injective or surjective.
{(a, b): a is a person, b is an ancestor of a}
Let the function f: R → R be defined by f(x) = cosx, ∀ x ∈ R. Show that f is neither one-one nor onto
Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:
h(x) = x|x|
Let f: R → R be defined by f(x) = `1/x` ∀ x ∈ R. Then f is ______.
Which of the following functions from Z into Z are bijections?
Let f : R → R be defind by f(x) = `1/"x" AA "x" in "R".` Then f is ____________.
Let f : R → R be a function defined by f(x) `= ("e"^abs"x" - "e"^-"x")/("e"^"x" + "e"^-"x")` then f(x) is
The function f: R → R defined as f(x) = x3 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.
- Let f: R → R be defined by f(x) = x − 4. Then the range of f(x) is ____________.
A function f: x → y is/are called onto (or surjective) if x under f.
Let the function f: R → R be defined by f(x) = 4x – 1, ∀ x ∈ R then 'f' is
The solution set of the inequation log1/3(x2 + x + 1) + 1 > 0 is ______.
`x^(log_5x) > 5` implies ______.
Let f: R→R be a polynomial function satisfying f(x + y) = f(x) + f(y) + 3xy(x + y) –1 ∀ x, y ∈ R and f'(0) = 1, then `lim_(x→∞)(f(2x))/(f(x)` is equal to ______.
If A = {x ∈ R: |x – 2| > 1}, B = `{x ∈ R : sqrt(x^2 - 3) > 1}`, C = {x ∈ R : |x – 4| ≥ 2} and Z is the set of all integers, then the number of subsets of the set (A ∩ B ∩ C) C ∩ Z is ______.
Which one of the following graphs is a function of x?
 |
 |
| Graph A | Graph B |
