Advertisements
Advertisements
Question
Let f : R `{- 4/3} `- 43 →">→ R be a function defined as f(x) = `(4x)/(3x +4)` . Show that f : R - `{-4/3}`→ Rang (f) is one-one and onto. Hence, find f -1.
Solution
The function f : R - `{-4/3}→ R - {4/3}` is given by f (x) = `(4x)/(3x +4).` Injectivity: Let `x,y in R - {-4/3} `be such that f (x) = f (y)
⇒ `(4x)/(3x + 4)= (4x)/(3y+4)`
⇒ 4x (3y +4) = 4y (3x +4)
⇒ 12xy + 16x = 12xy +16y
⇒ 16x = 16y
⇒ x = y
Hence, f is one-one function.
Surjectivity: Let y be an arbitrary element of `R - {4/3}.` Then,
f(x) = y
⇒ `(4x)/(3x+4) = y`
⇒ 4x = 3xy +4y
⇒ 4x - 3xy = 4y
⇒ `x = (4y)/(4-3y)`
As `y ∈ R -{4/3} , (4y)/(4-3y) in R.`
Also , `(4y)/(4 - 3y) ≠ -4/3` because `(4y)/(4-3y) = - 4/3 ⇒ 12y = -16 +12y ⇒ 0 = -16,`which is not posssible.
Thus,
`x = (4y)/(4-3y) in R - {- 4/3}` sich that
`f (x) = f ((4x)/(3x +4))= (4((4y)/(4 -3y)))/(3((4y)/(4 -3y))+4) = (16y)/(12y+ 16 - 12y)= (16y)/16` = y , so every element in `R - {4/3} ` has pre-image in `R- {-4/3}.`
Hence, f is onto.
Now,
`x = (4y)/(4 -3y)`
Replacing x by f-1 and y by x, we have
`f^-1 (x) = (4x)/(4 - 3x)`
APPEARS IN
RELATED QUESTIONS
Check the injectivity and surjectivity of the following function:
f: N → N given by f(x) = x2
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.
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
Show that function f: R `rightarrow` {x ∈ R : −1 < x < 1} defined by f(x) = `x/(1 + |x|)`, x ∈ R is one-one and onto function.
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, 2), (b, 1), (c, 1)}
Let f = {(1, −1), (4, −2), (9, −3), (16, 4)} and g = {(−1, −2), (−2, −4), (−3, −6), (4, 8)}. Show that gof is defined while fog is not defined. Also, find gof.
Find fog (2) and gof (1) when : f : R → R ; f(x) = x2 + 8 and g : R → R; g(x) = 3x3 + 1.
Find fog and gof if : f (x) = |x|, g (x) = sin x .
Find fog and gof if : f(x) = c, c ∈ R, g(x) = sin `x^2`
If f(x) = 2x + 5 and g(x) = x2 + 1 be two real functions, then describe each of the following functions:
(1) fog
(2) gof
(3) fof
(4) f2
Also, show that fof ≠ f2
if f (x) = `sqrt (x +3) and g (x) = x ^2 + 1` be two real functions, then find fog and gof.
If f : Q → Q, g : Q → Q are two functions defined by f(x) = 2 x and g(x) = x + 2, show that f and g are bijective maps. Verify that (gof)−1 = f−1 og −1.
Let A = R - {3} and B = R - {1}. Consider the function f : A → B defined by f(x) = `(x-2)/(x-3).`Show that f is one-one and onto and hence find f-1.
[CBSE 2012, 2014]
If f : R → (0, 2) defined by `f (x) =(e^x - e^(x))/(e^x +e^(-x))+1`is invertible , find f-1.
If A = {a, b, c} and B = {−2, −1, 0, 1, 2}, write the total number of one-one functions from A to B.
Let \[f : \left[ - \frac{\pi}{2}, \frac{\pi}{2} \right] \to\] A be defined by f(x) = sin x. If f is a bijection, write set A.
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]
Write the domain of the real function f defined by f(x) = `sqrt (25 -x^2)` [NCERT EXEMPLAR]
Let
\[A = \left\{ x \in R : - 1 \leq x \leq 1 \right\} = B\] Then, the mapping\[f : A \to \text{B given by} f\left( x \right) = x\left| x \right|\] is
Which of the following functions from
\[A = \left\{ x \in R : - 1 \leq x \leq 1 \right\}\]
If \[f : R \to \left( - 1, 1 \right)\] is defined by
\[f\left( x \right) = \frac{- x|x|}{1 + x^2}, \text{ then } f^{- 1} \left( x \right)\] equals
If \[g\left( x \right) = x^2 + x - 2\text{ and} \frac{1}{2} gof\left( x \right) = 2 x^2 - 5x + 2\] is equal to
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 − {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)
Set A has 3 elements and the set B has 4 elements. Then the number of injective mappings that can be defined from A to B is ______.
The domain of the function f: R → R defined by f(x) = `sqrt(x^2 - 3x + 2)` is ______
The smallest integer function f(x) = [x] is ____________.
The number of bijective functions from set A to itself when A contains 106 elements is ____________.
Let f : R → R be defind by f(x) = `1/"x" AA "x" in "R".` Then f is ____________.
Which of the following functions from Z into Z is bijective?
Range of `"f"("x") = sqrt((1 - "cos x") sqrt ((1 - "cos x")sqrt ((1 - "cos x")....infty))`
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 ____________.
A function f: x → y is/are called onto (or surjective) if x under f.
Let [x] denote the greatest integer ≤ x, where x ∈ R. If the domain of the real valued function f(x) = `sqrt((|[x]| - 2)/(|[x]| - 3)` is (–∞, a) ∪ [b, c) ∪ [4, ∞), a < b < c, then the value of a + b + c is ______.
The solution set of the inequation log1/3(x2 + x + 1) + 1 > 0 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)
Let S = {1, 2, 3, 4, 5, 6, 7}. Then the number of possible functions f: S `rightarrow` S such that f(m.n) = f(m).f(n) for every m, n ∈ S and m.n ∈ S is equal to ______.
Let f(x) be a polynomial function of degree 6 such that `d/dx (f(x))` = (x – 1)3 (x – 3)2, then
Assertion (A): f(x) has a minimum at x = 1.
Reason (R): When `d/dx (f(x)) < 0, ∀ x ∈ (a - h, a)` and `d/dx (f(x)) > 0, ∀ x ∈ (a, a + h)`; where 'h' is an infinitesimally small positive quantity, then f(x) has a minimum at x = a, provided f(x) is continuous at x = a.
