Question

Which is the inverse of the function a(d)=5d-3? And use the definition of inverse functions to prove a(d) and a-1(d) are inverse functions

Answer 1

`a'(d) = (d)/(5) + (3)/(5)`

`a(a'(d)) = a'(a(d)) = d`

Explanation:

Given

`a(d) = 5d - 3`

Solving (a): Write as inverse function

`a(d) = 5d - 3`

Represent a(d) as y

`y = 5d - 3`

Swap positions of d and y

`d = 5y - 3`

Make y the subject

`5y = d + 3`

`y = (d)/(5) + (3)/(5)`

Replace y with a'(d)

`a'(d) = (d)/(5) + (3)/(5)`

Prove that a(d) and a'(d) are inverse functions

`a'(d) = (d)/(5) + (3)/(5)` and
`a(d) = 5d - 3`

To do this, we prove that:

`a(a'(d)) = a'(a(d)) = d`

Solving for
`a(a'(d))`

`a(a'(d)) = a((d)/(5) + (3)/(5))`

Substitute
`(d)/(5) + (3)/(5)` for d in
`a(d) = 5d - 3`

`a(a'(d)) = 5((d)/(5) + (3)/(5)) - 3`

`a(a'(d)) = (5d)/(5) + (15)/(5) - 3`

`a(a'(d)) = d + 3 - 3`

`a(a'(d)) = d`

Solving for:
`a'(a(d))`

`a'(a(d)) = a'(5d - 3)`

Substitute 5d - 3 for d in
`a'(d) = (d)/(5) + (3)/(5)`

`a'(a(d)) = (5d - 3)/(5) + (3)/(5)`

Add fractions

`a'(a(d)) = (5d - 3+3)/(5)`

`a'(a(d)) = (5d)/(5)`

`a'(a(d)) = d`

Hence:

`a(a'(d)) = a'(a(d)) = d`