Question

For f(x) = 5x + 1

A. Find f(7).
B. Find f−1(x).
C. Find f−1(7).
D. Find f( f−1(7)).

Show all work

Answer 1

Answer with step-by-step explanation:

We are given the following function and we are to find f(7), f−1(x), f−1(7) and f( f−1(7)):

`f(x) = 5x + 1`

A. Find f(7):

`f(7)=5(7)+1 = 36`

B. Find f'1(x):

`y = 5x + 1`

Making x the subject to get:

`x = (y-1)/(5)`

`f'1(x) = (x-1)/(5)`

C. Find f−1(7):

`f'1(7) = \frac {7-1} {5} = \frac {6} {5}`

D. Find f( f−1(7)):

`x = f ( f^(-1) (7)) \\\\ f^(-1) (7) = \frac{6} {5} \\\\ x = f (\frac {6} {5} ) \\\\ f(\frac {6} {5}) = 5 (\frac {6} {5}) + 1 \\\\ f ( \frac {6} {5}) = 7`

Answer 2

A.
`f(7) = 5(7) + 1\\\\f(7) = 36`

B.
`f^(-1)(x) = (x - 1)/(5)`

C.
`f^(-1)(7) =(6)/(5)`

D.
`f((6)/(5)) = 7`

Explanation:

A. To solve the first part of the problem we must replace
`x = 7` in the function
`f(x) = 5x + 1`

So:

`f(7) = 5(7) + 1\\\\f(7) = 36`

B. In part B we must find the inverse function of
`f(x) = 5x + 1`

To find the inverse function do
`y = f(x)`

`y = 5x +1`

Now clear the variable x.

`(y - 1)/(5) = x`

Replace x with y.

`y = (x - 1)/(5)`

Finally

`f^(-1)(x) = (x - 1)/(5)`

C. Now we take the inverse function found above and replace
`x = 7`

`f^(-1)(7) = (7 - 1)/(5)\\\\f(7) = (6)/(5)`

D. Now we substitute
`x = f(f^(-1)(7))` in the original function.

`x = f( f^(-1)(7))\\\\f^(-1)(7) = (6)/(5)\\\\ x= f((6)/(5) )\\\\f((6)/(5)) = 5((6)/(5)) + 1\\\\f((6)/(5)) = 7`