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)`
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
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 in 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).
If the function `f(x) = sqrt(2x - 3)` is invertible then find its inverse. Hence prove that `(fof^(-1))(x) = x`
Let A = {−1, 0, 1} and f = {(x, x2) : x ∈ A}. Show that f : A → A is neither one-one nor onto.
Classify the following function as injection, surjection or bijection :
f : Z → Z, defined by f(x) = x − 5
Give examples of two surjective functions f1 and f2 from Z to Z such that f1 + f2 is not surjective.
Find fog and gof if : f(x) = sin−1 x, g(x) = x2
Let f be a real function given by f (x)=`sqrt (x-2)`
Find each of the following:
(i) fof
(ii) fofof
(iii) (fofof) (38)
(iv) f2
Also, show that fof ≠ `f^2` .
State with reason whether the following functions have inverse :
f : {1, 2, 3, 4} → {10} with f = {(1, 10), (2, 10), (3, 10), (4, 10)}
If f : C → C is defined by f(x) = x4, write f−1 (1).
If f : R → R is defined by f(x) = 10 x − 7, then write f−1 (x).
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]
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
A function f from the set of natural numbers to the set of integers defined by
\[f\left( n \right)\begin{cases}\frac{n - 1}{2}, & \text{when n is odd} \\ - \frac{n}{2}, & \text{when n is even}\end{cases}\]
Let \[f\left( x \right) = \frac{1}{1 - x} . \text{Then}, \left\{ f o \left( fof \right) \right\} \left( x \right)\]
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 \[f : R \to R\] is given by \[f\left( x \right) = x^3 + 3, \text{then} f^{- 1} \left( x \right)\] is equal to
Let A = R − (2) and B = R − (1). If f: A ⟶ B is a function defined by`"f(x)"=("x"-1)/("x"-2),` how that f is one-one and onto. Hence, find f−1.
Write about strlen() function.
Let f: R → R be the function defined by f(x) = 4x – 3 ∀ x ∈ R. Then write f–1
Let D be the domain of the real valued function f defined by f(x) = `sqrt(25 - x^2)`. Then, write D
Let C be the set of complex numbers. Prove that the mapping f: C → R given by f(z) = |z|, ∀ z ∈ C, 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|
Using the definition, prove that the function f: A→ B is invertible if and only if f is both one-one and onto
Let f: R – `{3/5}` → R be defined by f(x) = `(3x + 2)/(5x - 3)`. Then ______.
The number of bijective functions from set A to itself when A contains 106 elements 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 said to be one – one (or injective) if:
Function f: R → R, defined by f(x) = `x/(x^2 + 1)` ∀ x ∈ R is not
If f; R → R f(x) = 10x + 3 then f–1(x) is:
Consider a function f: `[0, pi/2] ->` R, given by f(x) = sinx and `g[0, pi/2] ->` R given by g(x) = cosx then f and g are
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 ______.
The solution set of the inequation log1/3(x2 + x + 1) + 1 > 0 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 ______.
The function f(x) = [x], where [x] denotes the greatest integer less than or equal to x; is continuous at ______.
Let A = R – {2} and B = R – {1}. If f: A `→` B is a function defined by f(x) = `(x - 1)/(x - 2)` then show that f is a one-one and an onto function.
If f : R `rightarrow` R is defined by `f(x) = (2x - 7)/4`, show that f(x) is one-one and onto.
