Advertisements
Advertisements
प्रश्न
If f : R → R is defined by f(x) = 3x − 5, prove that f is a bijection and find its inverse
Advertisements
उत्तर
Given f(x) = 3x – 5
Let y = 3x – 5
y + 5 = 3x
⇒ `(y + 5)/3` = x
Let g(y) = `(y + 5)/3`
gof(x) = g(f(x))
= g(3x – 5)
= `(3x - 5 + 5)/3`
= `(3x)/5`
= x
gof(x) = x
fog(y) = f(g(y))
= `f((y + 5)/3)`
= `3((y + 5)/3) - 5`
= y + 5 – 5
fog(y) = y
∴ gof = Ix and fog = IY
Hence f and g are bijections and inverses to each other.
Hence f is a bijection and f-1(y) = `(y + 5)/3`
Replacing y by x we get f-1(x) = `(x + 5)/3`
APPEARS IN
संबंधित प्रश्न
Write the values of f at − 4, 1, −2, 7, 0 if
f(x) = `{{:(- x + 4, "if" - ∞ < x ≤ - 3),(x + 4, "if" - 3 < x < -2),(x^2 - x, "if" - 2 ≤ x < 1),(x - x^2, "if" 1 ≤ x < 7),(0, "otherwise"):}`
Write the values of f at −3, 5, 2, −1, 0 if
f(x) = `{{:(x^2 + x - 5, "if" x ∈ (−∞, 0)),(x^2 + 3x - 2, "if" x ∈ (3, ∞)),(x^2, "if" x ∈ (0",", 2)),(x^2 - 3, "otherwise"):}`
State whether the following relations are functions or not. If it is a function check for one-to-oneness and ontoness. If it is not a function, state why?
If A = {a, b, c} and f = {(a, c), (b, c), (c, b)}; (f : A → A)
Let A = {1, 2, 3, 4} and B = {a, b, c, d}. Give a function from A → B of the following:
neither one-to-one nor onto
Let A = {1, 2, 3, 4} and B = {a, b, c, d}. Give a function from A → B of the following:
one-to-one but not onto
Let A = {1, 2, 3, 4} and B = {a, b, c, d}. Give a function from A → B of the following:
one-to-one and onto
Find the domain of `1/(1 - 2sinx)`
Find the largest possible domain of the real valued function f(x) = `sqrt(4 - x^2)/sqrt(x^2 - 9)`
Show that the relation xy = −2 is a function for a suitable domain. Find the domain and the range of the function
The weight of the muscles of a man is a function of his body weight x and can be expressed as W(x) = 0.35x. Determine the domain of this function
The distance of an object falling is a function of time t and can be expressed as s(t) = −16t2. Graph the function and determine if it is one-to-one.
The owner of a small restaurant can prepare a particular meal at a cost of Rupees 100. He estimates that if the menu price of the meal is x rupees, then the number of customers who will order that meal at that price in an evening is given by the function D(x) = 200 − x. Express his day revenue, total cost and profit on this meal as functions of x
Choose the correct alternative:
If f(x) = |x − 2| + |x + 2|, x ∈ R, then
Choose the correct alternative:
The range of the function f(x) = |[x] − x|, x ∈ R is
Choose the correct alternative:
The number of constant functions from a set containing m elements to a set containing n elements is
Choose the correct alternative:
The function f : [0, 2π] → [−1, 1] defined by f(x) = sin x is
Choose the correct alternative:
If the function f : [−3, 3] → S defined by f(x) = x2 is onto, then S is
The function f : R → R is defined by f(x) = `((x^2 + cos x)(1 + x^4))/((x - sin x)(2x - x^3)) + "e"^(-|x|)` is
