Question

State of the given functions are inverses

Answer 1

1) They are not inverses

2) They are inverses

Explanation:

We need to find the composition function between these functions to verify if these functions are inverses. If f[g(x)] and g[f(x)] are equal to x they are inverses.

1)

Let's find f[g(x)] and simplify.

`f[g(x)]=(1)/(2)g(x)+(3)/(2)`

`f[g(x)]=(1)/(2)(4-(3)/(2)x)+(3)/(2)`

`f[g(x)]=(7)/(2)-(3)/(4)x`

As f[g(x)] is not equal to x, these functions are not inverses.

2)

Let's find f[g(x)] and simplify.

`f[g(n)]=(-16+(4n+16))/(4)`

`f[g(n)]=(-16+4n+16)/(4)`

`f[g(n)]=(4n)/(4)`

`f[g(n)]=n`

Now, we need to find the other composition function g[f(x)]

Let's find g[f(x)] and simplify.

`g[f(x)]=4((-16+n)/(4))+16`

`g[f(x)]=-16+n+16`

`g[f(x)]=n`

Therefore, as f[g(n)] = g[f(n)] = n, both functions are inverses.

I hope it helps you!