Advertisements
Advertisements
Question
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.
Advertisements
Solution
Let A = R – {2}, B = R – {1}
f: A `→` B s.t. f(x) = `(x - 1)/(x - 2)`
For x1, x2 ∈ A
f(x1) = f(x2)
`\implies (x_1 - 1)/(x_1 - 2) = (x_2 - 1)/(x_2 - 2)`
`\implies (x_1 - 1)/(x_1 - 2) - 1 = (x_2 - 1)/(x_2 - 2) - 1`
`\implies 1/(x_1 - 2) = 1/(x_2 - 2)`
`\implies` x1 – 2 = x2 – 2
`\implies` x1 = x2
∴ f(x) is one-one function.
Also if f(x) = y, where y ∈ B.
`\implies (x - 1)/(x - 2)` = y
`\implies` x – 1 = xy – 2y
`\implies` 2y – 1 = xy – x
`\implies` 2y – 1 = x(y – 1)
x = `(2y - 1)/(y - 1) ∈ A`
Clearly every element y ∈ B is associated to x = `(2y - 1)/(y - 1)` of set A.
So Range of f = B `\implies` f is into
Hence f is one-one and onto function.
APPEARS IN
RELATED QUESTIONS
Classify the following function as injection, surjection or bijection : f : Z → Z given by f(x) = x2
Classify the following function as injection, surjection or bijection : f : Z → Z given by f(x) = x3
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = sinx
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = 3 − 4x
Show that the function f : R − {3} → R − {2} given by f(x) = `(x-2)/(x-3)` is a bijection.
Let A = [-1, 1]. Then, discuss whether the following function from A to itself is one-one, onto or bijective : g(x) = |x|
Verify associativity for the following three mappings : f : N → Z0 (the set of non-zero integers), g : Z0 → Q and h : Q → R given by f(x) = 2x, g(x) = 1/x and h(x) = ex.
Let f(x) = x2 + x + 1 and g(x) = sin x. Show that fog ≠ gof.
If f, g : R → R be two functions defined as f(x) = |x| + x and g(x) = |x|- x, ∀x∈R" .Then find fog and gof. Hence find fog(–3), fog(5) and gof (–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)}
Consider f : R → R+ → [4, ∞) given by f(x) = x2 + 4. Show that f is invertible with inverse f−1 of f given by f−1 `(x)= sqrt (x-4)` where R+ is the set of all non-negative real numbers.
If f : C → C is defined by f(x) = x2, write f−1 (−4). Here, C denotes the set of all complex numbers.
Let f : R → R be the function defined by f(x) = 4x − 3 for all x ∈ R Then write f . [NCERT EXEMPLAR]
\[f : R \to R \text{given by} f\left( x \right) = x + \sqrt{x^2} \text{ is }\]
The function \[f : [0, \infty ) \to \text {R given by } f\left( x \right) = \frac{x}{x + 1} is\]
Let
Let [x] denote the greatest integer less than or equal to x. If \[f\left( x \right) = \sin^{- 1} x, g\left( x \right) = \left[ x^2 \right]\text{ and } h\left( x \right) = 2x, \frac{1}{2} \leq x \leq \frac{1}{\sqrt{2}}\]
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
Consider the set A containing n elements. Then, the total number of injective functions from A onto itself is ______
Using the definition, prove that the function f: A→ B is invertible if and only if f is both one-one and onto
The mapping f : N → N is given by f(n) = 1 + n2, n ∈ N when N is the set of natural numbers is ____________.
Let f : R `->` R be a function defined by f(x) = x3 + 4, 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.
- The function f: Z → Z defined by f(x) = x2 is ____________.
If f: R → R given by f(x) =(3 − x3)1/3, find f0f(x)
Function f: R → R, defined by f(x) = `x/(x^2 + 1)` ∀ x ∈ R is not
Number of integral values of x satisfying the inequality `(3/4)^(6x + 10 - x^2) < 27/64` is ______.
Let f: R→R be a polynomial function satisfying f(x + y) = f(x) + f(y) + 3xy(x + y) –1 ∀ x, y ∈ R and f'(0) = 1, then `lim_(x→∞)(f(2x))/(f(x)` is equal to ______.
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)
The graph of the function y = f(x) is symmetrical about the line x = 2, then ______.
