Advertisements
Advertisements
प्रश्न
Mark the correct alternative in the following question:
Let f : R \[-\] \[\left\{ \frac{3}{5} \right\}\] \[\to\] R be defined by f(x) = \[\frac{3x + 2}{5x - 3}\] Then,
विकल्प
f-1 (x) = f (x)
`f^-1 (x) = - f(x)`
fo f(x) = - x
`f^-1(x) = 1/19f(x)`
Advertisements
उत्तर
We have,
f : R \[-\] \[\left\{ \frac{3}{5} \right\}\] \[\to\] R be defined by f(x) = \[\frac{3x + 2}{5x - 3}\]
\[fof\left( x \right) = f\left( f\left( x \right) \right)\]
\[ = f\left( \frac{3x + 2}{5x - 3} \right)\]
\[ = \frac{3\left( \frac{3x + 2}{5x - 3} \right) + 2}{5\left( \frac{3x + 2}{5x - 3} \right) - 3}\]
\[ = \frac{\left( \frac{9x + 6}{5x - 3} \right) + 2}{\left( \frac{15x + 10}{5x - 3} \right) - 3}\]
\[ = \frac{\left( \frac{9x + 6 + 10x - 6}{5x - 3} \right)}{\left( \frac{15x + 10 - 15x + 9}{5x - 3} \right)}\]
\[ = \frac{19x}{19}\]
\[ = x\]
\[\text{Let } y = \frac{3x + 2}{5x - 3}\]
\[ \Rightarrow 5xy - 3y = 3x + 2\]
\[ \Rightarrow 5xy - 3x = 3y + 2\]
\[ \Rightarrow x\left( 5y - 3 \right) = 3y + 2\]
\[ \Rightarrow x = \frac{3y + 2}{5y - 3}\]
\[ \Rightarrow f^{- 1} \left( y \right) = \frac{3y + 2}{5y - 3}\]
\[So, f^{- 1} \left( x \right) = \frac{3x + 2}{5x - 3} = f\left( x \right)\]
Hence, the correct alternative is option (a).
APPEARS IN
संबंधित प्रश्न
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
Find the number of all onto functions from the set {1, 2, 3, ..., n} to itself.
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)}
Let A = {–1, 0, 1, 2}, B = {–4, –2, 0, 2} and f, g : A → B be functions defined by f(x) = x2 – x, x ∈ A and g(x) = `2|x - 1/2| – 1`, x ∈ A. Are f and g equal?
Justify your answer. (Hint: One may note that two functions f : A → B and g : A → B such that f(a) = g(a) ∀ a ∈ A are called equal functions.)
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)
Give an example of a function which is one-one but not onto ?
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = sin2x + cos2x
Set of ordered pair of a function? If so, examine whether the mapping is injective or surjective :{(x, y) : x is a person, y is the mother of x}
If f : R → R be the function defined by f(x) = 4x3 + 7, show that f is a bijection.
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.
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.
Find fog and gof if : f(x) = c, c ∈ R, g(x) = sin `x^2`
If f(x) = sin x and g(x) = 2x be two real functions, then describe gof and fog. Are these equal functions?
Let f : R → R+ be defined by f(x) = ax, a > 0 and a ≠ 1. Write f−1 (x).
Let `f : R - {- 3/5}` → R be a function defined as `f (x) = (2x)/(5x +3).`
f-1 : Range of f → `R -{-3/5}`.
Let f, g : R → R be defined by f(x) = 2x + l and g(x) = x2−2 for all x
∈ R, respectively. Then, find gof. [NCERT EXEMPLAR]
Let
f : R → R be given by
\[f\left( x \right) = \left[ x^2 \right] + \left[ x + 1 \right] - 3\]
where [x] denotes the greatest integer less than or equal to x. Then, f(x) is
(d) one-one and onto
Which of the following functions from
to itself are bijections?
\[f : Z \to Z\] be given by
` f (x) = {(x/2, ", if x is even" ) ,(0 , ", if x is odd "):}`
Then, f is
Mark the correct alternative in the following question:
If the set A contains 7 elements and the set B contains 10 elements, then the number one-one functions from A to B is
Show that the function f: R → R defined by f(x) = `x/(x^2 + 1)`, ∀ ∈ + R , is neither one-one nor onto
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 X = {1, 2, 3}and Y = {4, 5}. Find whether the following subset of X ×Y are function from X to Y or not
f = {(1, 4), (1, 5), (2, 4), (3, 5)}
Let X = {1, 2, 3}and Y = {4, 5}. Find whether the following subset of X ×Y are function from X to Y or not
h = {(1,4), (2, 5), (3, 5)}
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 ______.
Which of the following functions from Z into Z are bijections?
Let f: `[2, oo)` → R be the function defined by f(x) = x2 – 4x + 5, then the range of f is ______.
The mapping f : N → N is given by f(n) = 1 + n2, n ∈ N when N is the set of natural numbers 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 ____________.
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 ____________.
If f: R → R given by f(x) =(3 − x3)1/3, find f0f(x)
The domain of the function `cos^-1((2sin^-1(1/(4x^2-1)))/π)` is ______.
If log102 = 0.3010.log103 = 0.4771 then the number of ciphers after decimal before a significant figure comes in `(5/3)^-100` is ______.
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)`.
If f : R `rightarrow` R is defined by `f(x) = (2x - 7)/4`, show that f(x) is one-one and onto.
A function is called one-one if different elements of the domain have:
A function is onto when every element of the codomain has:
If every seat in a classroom is occupied by one student, the situation resembles:
