Question

Consider given by , show that f is invertible with f-1(y) =((√(y+6)-1)/3) . Hence find (i) (10)(ii) y if (y) = 4/3 where R, is the set of all non-negative real numbers.

Answer 1

Given:

A function  f-1(y) =((√(y+6)-1)/3) .

To Find:

Show that f is invertible.

(i) f(10)

(ii) y if (y) = 4/3

Solution:

Since R, is the set of all non-negative real numbers,

• y + 6 can only positive value.

• √(y+6) will be unique positive number.

• Hence f-1(y) is one - one .

• All the values of f-(y) > 0 , as min(√(y+6)) = √6 > 1
• Therefore f-1(y) is Onto.
• Since f-1(y) is one one and onto , f-1(y) is invertible and hence f(y) is also invertible.

• Its given that with f-1(y) =((√(y+6)-1)/3) .

Therefore , we need to find y which can be substituted to f(y) .

• 3f-1(y) = √(y+6) - 1
• 3f-1(y) + 1 = √ ( y + 6)
• (3f-1(y) + 1)² = y + 6
• (3f-1(y) + 1 )² - 6 = y

Therefore,

• f(y) = (3y + 1)² - 6

1. f ( 10 )

• f ( 10 ) = ( 3x10 + 1)² - 6
• f ( 10 ) = 31² - 6 = 961 - 6 = 955

    2. y , if f (y) = 4/3

• given that with f-1(y) =((√(y+6)-1)/3)
• y = ( ( √(4/3 + 6) - 1 )/3)
• y = ( ( √22/3  - 1 )/3)

Therefore f(10) = 955 and y when f-1(y) = 4/3 is  ( √22 - √3 )/3√3).

Answer 2

sorry friend

Step-by-step explanation:

l didn't know answer this question