Advertisements
Advertisements
प्रश्न
Let A = ℝ − {3}, B = ℝ − {1}. Let f : A → B be defined by \[f\left( x \right) = \frac{x - 2}{x - 3}, \forall x \in A\] Show that f is bijective. Also, find
(i) x, if f−1(x) = 4
(ii) f−1(7)
Advertisements
उत्तर
We have,
A = R \[-\] {3} and B = R \[-\] {1}
The function f : A \[\to\] B defined by f(x) = \[\frac{x - 2}{x - 3}\]
\[\text { Let }x, y \text { in A such that }f\left( x \right) = f\left( y \right) . \text { Then }, \]
\[\frac{x - 2}{x - 3} = \frac{y - 2}{y - 3}\]
\[ \Rightarrow xy - 3x - 2y + 6 = xy - 2x - 3y + 6\]
\[ \Rightarrow - x = - y\]
\[ \Rightarrow x = y\]
\[ \therefore\text { f is one - one } .\]
\[\text { Let y } \in B . \text { Then, y } \neq 1 . \]
\[\text { The function f is onto if there exists } x \text { in A such that f }\left( x \right) = y . \]
\[\text {Now,}\]
\[f\left( x \right) = y\]
\[ \Rightarrow \frac{x - 2}{x - 3} = y\]
\[ \Rightarrow x - 2 = xy - 3y\]
\[ \Rightarrow x - xy = 2 - 3y\]
\[ \Rightarrow x\left( 1 - y \right) = 2 - 3y\]
\[ \Rightarrow x = \frac{2 - 3y}{1 - y} \in A \left[ y \neq 1 \right]\]
\[\text { Thus, for any } y \text { in B, there exists } \frac{2 - 3y}{1 - y} \text { in A such that }\]
\[f\left( \frac{2 - 3y}{1 - y} \right) = \frac{\left( \frac{2 - 3y}{1 - y} \right) - 2}{\left( \frac{2 - 3y}{1 - y} \right) - 3} = \frac{2 - 3y - 2 + 2y}{2 - 3y - 3 + 3y} = \frac{- y}{- 1} = y\]
\[ \therefore \text { f is onto } .\]
\[\text { So, f is one - one and onto fucntion } . \]
\[\text { Now }, \]
\[\text { As,} x = \left( \frac{2 - 3y}{1 - y} \right)\]
\[\text { So }, f^{- 1} \left( x \right) = \left( \frac{2 - 3x}{1 - x} \right) = \frac{3x - 2}{x - 1}\]
Therefore,f is bijective.
(i) \[f^{- 1} \left( x \right) = 4\]
\[ \Rightarrow \frac{3x - 2}{x - 1} = 4\]
\[ \Rightarrow 3x - 2 = 4x - 4\]
\[ \Rightarrow x = 2\]
\[f^{- 1} \left( x \right) = \frac{3x - 2}{x - 1}\]
\[ \Rightarrow f^{- 1} \left( 7 \right) = \frac{3\left( 7 \right) - 2}{7 - 1} = \frac{19}{6}\]
APPEARS IN
संबंधित प्रश्न
Check the injectivity and surjectivity of the following function:
f : N → N 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.
Show that the signum function f : R → R, given by
`f(x) = {(1", if" x > 0), (0", if" x = 0), (-1", if" x < 0):}`
is neither one-one nor onto.
Let f: R → R be the Signum Function defined as
f(x) = `{(1,x>0), (0, x =0),(-1, x< 0):}`
and g: R → R be the Greatest Integer Function given by g(x) = [x], where [x] is greatest integer less than or equal to x. Then does fog and gof coincide in (0, 1]?
Classify the following function as injection, surjection or bijection :
f : Q → Q, defined by f(x) = x3 + 1
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = `x/(x^2 +1)`
Show that the logarithmic function f : R0+ → R given by f (x) loga x ,a> 0 is a bijection.
If A = {1, 2, 3}, show that a onto function f : A → A must be one-one.
Let
f (x) =`{ (1 + x, 0≤ x ≤ 2) , (3 -x , 2 < x ≤ 3):}`
Find fof.
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)}
Let A and B be two sets, each with a finite number of elements. Assume that there is an injective map from A to B and that there is an injective map from B to A. Prove that there is a bijection from A to B.
If f : A → A, g : A → A are two bijections, then prove that fog is an injection ?
If f : R → R is defined by f(x) = x2, find f−1 (−25).
Let f : R → R be defined as `f (x) = (2x - 3)/4.` write fo f-1 (1) .
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)`.
Write whether f : R → R, given by `f(x) = x + sqrtx^2` is one-one, many-one, onto or into.
If f : {5, 6} → {2, 3} and g : {2, 3} → {5, 6} are given by f = {(5, 2), (6, 3)} and g = {(2, 5), (3, 6)}, then find fog. [NCERT EXEMPLAR]
Mark the correct alternative in the following question:
Let f : R→ R be defined as, f(x) = \[\begin{cases}2x, if x > 3 \\ x^2 , if 1 < x \leq 3 \\ 3x, if x \leq 1\end{cases}\]
Then, find f( \[-\]1) + f(2) + f(4)
Write about strlen() function.
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 f: R → R be defined by f(x) = `1/x` ∀ x ∈ R. Then f is ______.
Raji visited the Exhibition along with her family. The Exhibition had a huge swing, which attracted many children. Raji found that the swing traced the path of a Parabola as given by y = x2.
Answer the following questions using the above information.
- Let f: {1,2,3,....} → {1,4,9,....} be defined by f(x) = x2 is ____________.
Let f: R → R defined by f(x) = 3x. Choose the correct answer
A function f: x → y is/are called onto (or surjective) if x under f.
Function f: R → R, defined by f(x) = `x/(x^2 + 1)` ∀ x ∈ R is not
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 a and b are two positive integers such that b ≠ 1. Let g(a, b) = Number of lattice points inside the quadrilateral formed by lines x = 0, y = 0, x = b and y = a. f(a, b) = `[a/b] + [(2a)/b] + ... + [((b - 1)a)/b]`, then the value of `[(g(101, 37))/(f(101, 37))]` is ______.
(Note P(x, y) is lattice point if x, y ∈ I)
(where [.] denotes greatest integer function)
If f : R `rightarrow` R is defined by `f(x) = (2x - 7)/4`, show that f(x) is one-one and onto.
The function defined by \[\mathrm{f}(x)=\frac{2x+3}{3x+4},x\neq-\frac{4}{3}\] is
