Advertisements
Advertisements
प्रश्न
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = 5x3 + 4
Advertisements
उत्तर
f : R → R, defined by f(x) = 5x3 + 4
Injection test :
Let x and y be any two elements in the domain (R), such that f(x) = f(y).
f(x) = f(y)
5x3+4 = 5y3+4
5x3= 5y3
x3= y3
x = y
So, f is 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
5x3+4 = y
5x3+ 4 = y
`x^3 = (y - 4)/5 ∈ R `
So, f is a surjection and f is a bijection.
APPEARS IN
संबंधित प्रश्न
Prove that the greatest integer function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.
Let S = {a, b, c} and T = {1, 2, 3}. Find F−1 of the following functions F from S to T, if it exists.
F = {(a, 3), (b, 2), (c, 1)}
Give an example of a function which is not one-one but onto ?
Classify the following function as injection, surjection or bijection : f : Z → Z given by f(x) = x3
Classify the following function as injection, surjection or bijection :
f : Z → Z, defined by f(x) = x2 + x
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.
If f : A → B and g : B → C are one-one functions, show that gof is a one-one function.
Find fog and gof if : f (x) = |x|, g (x) = sin x .
Find fog and gof if : f(x) = sin−1 x, g(x) = x2
` if f : (-π/2 , π/2)` → R and g : [−1, 1]→ R be defined as f(x) = tan x and g(x) = `sqrt(1 - x^2)` respectively, describe fog and gof.
Let f : [−1, ∞) → [−1, ∞) be given by f(x) = (x + 1)2 − 1, x ≥ −1. Show that f is invertible. Also, find the set S = {x : f(x) = f−1 (x)}.
Let f be a function from R to R, such that f(x) = cos (x + 2). Is f invertible? Justify your answer.
If f : C → C is defined by f(x) = x2, write f−1 (−4). Here, C denotes the set of all complex numbers.
If f : C → C is defined by f(x) = x4, write f−1 (1).
If f : C → C is defined by f(x) = (x − 2)3, write f−1 (−1).
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).
Write whether f : R → R, given by `f(x) = x + sqrtx^2` is one-one, many-one, onto or into.
Let
\[A = \left\{ x : - 1 \leq x \leq 1 \right\} \text{and} f : A \to \text{A such that f}\left( x \right) = x|x|\]
Let
\[f : R - \left\{ n \right\} \to R\]
Let \[f\left( x \right) = x^2 and g\left( x \right) = 2^x\] Then, the solution set of the equation
If \[g \left( f \left( x \right) \right) = \left| \sin x \right| \text{and} f \left( g \left( x \right) \right) = \left( \sin \sqrt{x} \right)^2 , \text{then}\]
Let [x] denote the greatest integer less than or equal to x. If \[f\left( x \right) = \sin^{- 1} x, g\left( x \right) = \left[ x^2 \right]\text{ and } h\left( x \right) = 2x, \frac{1}{2} \leq x \leq \frac{1}{\sqrt{2}}\]
Let \[f\left(x\right) = x^3\] be a function with domain {0, 1, 2, 3}. Then domain of \[f^{-1}\] is ______.
If f(x) = `(x+3)/(4x−5) , "g"(x) = (3+5x)/(4x−1)` then verify that `("fog") (x)` = x.
Show that the function f: R → R defined by f(x) = `x/(x^2 + 1)`, ∀ ∈ + R , is neither one-one nor onto
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 f: R → R be defined by f(x) = x2 + 1. Then, pre-images of 17 and – 3, respectively, are ______.
Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:
g(x) = |x|
If the set A contains 5 elements and the set B contains 6 elements, then the number of one-one and onto mappings from A to B is ______.
Let f: R → R be given by f(x) = tan x. Then f–1(1) is ______.
The number of bijective functions from set A to itself when A contains 106 elements is ____________.
Let f : [0, ∞) → [0, 2] be defined by `"f" ("x") = (2"x")/(1 + "x"),` then f is ____________.
Let f : R `->` R be a function defined by f(x) = x3 + 4, then f is ______.
An organization conducted a bike race under 2 different categories-boys and girls. Totally there were 250 participants. Among all of them finally, three from Category 1 and two from Category 2 were selected for the final race. Ravi forms two sets B and G with these participants for his college project. Let B = {b1,b2,b3} G={g1,g2} where B represents the set of boys selected and G the set of girls who were selected for the final race.
Ravi decides to explore these sets for various types of relations and functions.
- Let R: B → G be defined by R = { (b1,g1), (b2,g2),(b3,g1)}, then R is ____________.
Consider a set containing function A= {cos–1cosx, sin(sin–1x), sinx((sinx)2 – 1), etan{x}, `e^(|cosx| + |sinx|)`, sin(tan(cosx)), sin(tanx)}. B, C, D, are subsets of A, such that B contains periodic functions, C contains even functions, D contains odd functions then the value of n(B ∩ C) + n(B ∩ D) is ______ where {.} denotes the fractional part of functions)
Which one of the following graphs is a function of x?
 |
 |
| Graph A | Graph B |
The trigonometric equation tan–1x = 3tan–1 a has solution for ______.
