Advertisements
Advertisements
प्रश्न
Show that the function f: ℝ → ℝ defined by f(x) = `x/(x^2 + 1), ∀x in R`is neither one-one nor onto. Also, if g: ℝ → ℝ is defined as g(x) = 2x - 1. Find fog(x)
Advertisements
उत्तर
Given `y = x/(x^2+1)`
`=> yx^2 - x + y = 0`
Here a = y, b = -1 and c = y
`:. x = (-(-1)+- sqrt(1-4y^2))/(2y)`
Clearly for every value of y, x will have two different values so the function is many−one not one−one
Since `1 -4y^2 >= 0 => (1+2y)(1-2y)>= 0 => (-1)/2 <= y ><= 1/2`
That means no matter what is x, y always belongs to the interval `[(-1)/2, 1/2]`
So, the function is not onto
Now, fog(x) = `(2x-1)/((2x-1)^2 +1) = (2x+1)/(4x^2 - 4x + 1+1) = (2x+1)/(2(2x^2 - 2x + 1))`
APPEARS IN
संबंधित प्रश्न
Check the injectivity and surjectivity of the following function:
f : R → R given by f(x) = x2
Show that the modulus function f : R → R, given by f(x) = |x|, is neither one-one nor onto, where |x| is x, if x is positive or 0 and |x| is –x, if x is negative.
In the following case, state whether the function is one-one, onto or bijective. Justify your answer.
f : R → R defined by f(x) = 1 + x2
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.
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = x3 − x
Let A = {1, 2, 3}. Write all one-one from A to itself.
Find gof and fog when f : R → R and g : R → R is defined by f(x) = 2x + 3 and g(x) = x2 + 5 .
Find fog and gof if : f (x) = x+1, g(x) = `e^x`
.
If f : R → (−1, 1) defined by `f (x) = (10^x- 10^-x)/(10^x + 10 ^-x)` is invertible, find f−1.
If f : R → R is given by f(x) = x3, write f−1 (1).
If f : R → R is defined by f(x) = 10 x − 7, then write f−1 (x).
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) = sqrt([x] - x) .`
Write the domain of the real function
`f (x) = 1/(sqrt([x] - x)`.
If f(x) = x + 7 and g(x) = x − 7, x ∈ R, write fog (7).
The range of the function
\[f\left( x \right) =^{7 - x} P_{x - 3}\]
Which of the following functions from
\[A = \left\{ x \in R : - 1 \leq x \leq 1 \right\}\]
Let
\[f : R \to R\] be given by \[f\left( x \right) = x^2 - 3\] Then, \[f^{- 1}\] is given by
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 defined by f(x) = x2 + 1. Then, pre-images of 17 and – 3, respectively, are ______.
For sets A, B and C, let f: A → B, g: B → C be functions such that g o f is injective. Then both f and g are injective functions.
Let A = R – {3}, B = R – {1}. Let f: A → B be defined by f(x) = `(x - 2)/(x - 3)` ∀ x ∈ A . Then show that f is bijective.
Using the definition, prove that the function f: A→ B is invertible if and only if f is both one-one and onto
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 ______.
The smallest integer function f(x) = [x] is ____________.
A general election of Lok Sabha is a gigantic exercise. About 911 million people were eligible to vote and voter turnout was about 67%, the highest ever
Let I be the set of all citizens of India who were eligible to exercise their voting right in the general election held in 2019. A relation ‘R’ is defined on I as follows:
R = {(V1, V2) ∶ V1, V2 ∈ I and both use their voting right in the general election - 2019}
- Mr. ’X’ and his wife ‘W’ both exercised their voting right in the general election-2019, Which of the following is true?
A general election of Lok Sabha is a gigantic exercise. About 911 million people were eligible to vote and voter turnout was about 67%, the highest ever
Let I be the set of all citizens of India who were eligible to exercise their voting right in the general election held in 2019. A relation ‘R’ is defined on I as follows:
R = {(V1, V2) ∶ V1, V2 ∈ I and both use their voting right in the general election - 2019}
- Three friends F1, F2, and F3 exercised their voting right in general election-2019, then which of the following is true?
If f: R→R is a function defined by f(x) = `[x - 1]cos((2x - 1)/2)π`, where [ ] denotes the greatest integer function, then f 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 ______.
A function f : [– 4, 4] `rightarrow` [0, 4] is given by f(x) = `sqrt(16 - x^2)`. Show that f is an onto function but not a one-one function. Further, find all possible values of 'a' for which f(a) = `sqrt(7)`.
