Advertisements
Advertisements
Question
The formula for converting from Fahrenheit to Celsius temperatures is y = `(5x)/9 - 160/9`. Find the inverse of this function and determine whether the inverse is also a function
Solution
Let f(x) = `(5x)/9 - 160/9`
= `(5x - 160)/9`
Given y = `(5x)/9 - 160/9`
= `(5x - 160)/9`
Then 9y = 5x – 160
⇒ 5x = 9y + 160
⇒ x = `(9y + 160)/5`
Let g(y) = `(9y + 160)/5`
g o f(x) = g(f(x))
= `g((5x - 160)/9)`
= `(9((5x - 160)/9) +160)/5`
= `(5x - 160 + 60)/5`
= `(5x)/5`
g o f(x) = x
f o g(y) = f(g(y))
= `f((9y + 160)/5)`
= `(5((9y + 160)/5) - 160)/9`
= `(9y + 160 - 160)/9`
= `(9y)/9`
f o g(y) = y
Thus g o f =Ix nd f o g = Iy
This shows that f and g are bijections and inverses of each other.
f–1(y = `(9y + 160)/5`
Replacing y by x we get f–1(x) = `(9x + 160)/5`
= `(9x)/5 + 32`
APPEARS IN
RELATED QUESTIONS
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:
not one-to-one but 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
Find the range of the function `1/(2 cos x - 1)`
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.
A salesperson whose annual earnings can be represented by the function A(x) = 30,000 + 0.04x, where x is the rupee value of the merchandise he sells. His son is also in sales and his earnings are represented by the function S(x) = 25,000 + 0.05x. Find (A + S)(x) and determine the total family income if they each sell Rupees 1,50,00,000 worth of merchandise
The function for exchanging American dollars for Singapore Dollar on a given day is f(x) = 1.23x, where x represents the number of American dollars. On the same day the function for exchanging Singapore Dollar to Indian Rupee is g(y) = 50.50y, where y represents the number of Singapore dollars. Write a function which will give the exchange rate of American dollars in terms of Indian rupee
Choose the correct alternative:
If f(x) = |x − 2| + |x + 2|, x ∈ R, then
Choose the correct alternative:
The range of the function `1/(1 - 2 sin x)` 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:
If the function f : [−3, 3] → S defined by f(x) = x2 is onto, then S is
Choose the correct alternative:
The inverse of f(x) = `{{:(x, "if" x < 1),(x^2, "if" 1 ≤ x ≤ 4),(8sqrt(x), "if" x > 4):}` is
Choose the correct alternative:
The function f : R → R is defined by f(x) = sin x + cos x is
