Please help me with this problem. For each pair of functions f and g below, find f(g(x)) and g(f(x)).  Then, determine whether f and g are inverses of each other.     f(x) = 3x + 1     g(x) = 3x - 1     f(g(x)) =     g(f(x)) =

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`f(x) =3x +1`

`g(x)=3x -1`

I. `f(g(x))=?`

To solve this, consider the function f.

`f(x) = 3x + 1`

Then, replace the x with g(x).

`f(g(x))= 3g(x) + 1`

Plug-in `g(x) = 3x - 1` .

`f(g(x))=3(3x-1) + 1`

`f(g(x))= 9x - 3 + 1`

`f(g(x)) = 9x - 2`

...

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`f(x) =3x +1`

`g(x)=3x -1`

I. `f(g(x))=?`

To solve this, consider the function f.

`f(x) = 3x + 1`

Then, replace the x with g(x).

`f(g(x))= 3g(x) + 1`

Plug-in `g(x) = 3x - 1` .

`f(g(x))=3(3x-1) + 1`

`f(g(x))= 9x - 3 + 1`

`f(g(x)) = 9x - 2`

Therefore,  `f(g(x))=9x - 2 ` .

II.` g(f(x))=?`

To solve this, consider the function g.

`g(x)=3x-1`

Then, replace the x with f(x).

`g(f(x))=3f(x)-1`

Plug-in `f(x)=3x+1` .

`g(f(x))=3(3x+1) -1`

`g(f(x))=9x+3-1`

`g(f(x))=9x + 2`

Thus, `g(f(x))=9x +2` .

III. Verify if f(x) and g(x) are inverse of each other.

Take note that two functions f(x) and g(x) are inverse of each other if both composite function f(g(x)) and g(f(x)) are equal to x.

`f(g(x))=g(f(x))=x`

Since the composite functions above are:

`f(g(x))=9x-2`

and

`g(f(x))=9x+2`

therefore `f(x)=3x+1` and `g(x)=3x-1` are not inverse of each other.

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For f(g(x)), you just replace `g(x) = 3x - 1` in place of x in `f(x) = 3x + 1`
So,
`f(g(x)) = f(3x - 1)`
`f(g(x)) = 3(3x - 1) + 1`

Simplify the expression by distributing 3 into (3x - 1).
`f(g(x)) = 9x - 3 + 1`

Combine similar terms
`f(g(x)) = 9x - 2`

The same goes for g(f(x))

`g(f(x)) = g(3x + 1) - 1`
`g(f(x)) = 3(3x + 1) - 1`
`g(f(x)) = 9x + 3 - 1`
`g(f(x)) = 9x + 2`

Thus, the answers are
`f(g(x)) = 9x - 2` and
`g(f(x)) = 9x + 2`

Approved by eNotes Editorial Team