double functions

`f  g  (x)` = Composite function (`x`)

Always work right hand side to left. This means you take the entire machine `g  (x)` and plug it into machine `f  (x)` everywhere where there is an input `x`.

1 a.  `g  (x) = x + 4`

Then this machine "`g`" takes the input `x` and adds 4.

1 b.  `f  (x) = 2x^2``-3x - 5`

This machine `f` takes the input `x` and squares it, multiplies by 2, it then multiplies the input `x` by 3 and takes it away, it then takes away 5.

1 c.  `f  g  (x)`

From the right hand side take `g  (x)` and plug it into `f  (x)` everywhere there is an input `x`.

2 a.  `g  (x) = (x - 1)^2`  and   `h  (x) = 1 — 2 x`

Work out `gh  (5)`

To do this start from the right hand side. 

2 b.  `h  (x) = 1 - 2 x`

`h  (5)` means this machine "`h`", takes the input 5 multiplies it by 2 and takes it away from 1.

`h  (5) = 1 - 2 (5) = 1 - 10 = -9`

2 c.  `gh  (5)`

This means this machine "`g`" uses the input `h  (5)` which we know is -9.

`g  (- 9) = (x - 1)^2`

`g  (- 9) = (-9 - 1)^2`

`g  (- 9) = (-10)^2 = 100`

Answer: `g  h  (5) = 100`

3.  `f  (x) = x^2`  and  `g  (x) = x + 1`

Find `f  g  (x)` 

From the right hand side take `g  (x)` and plug it into `f  (x)` everywhere where there is an input `x`.

Answer `f  g  (x) = (x + 1)^2`