Composite functions - several functions together

Functions can be denoted as an `f` or `g`  or `h`  or any letter. Therefore you might see something like `gf(x)`.

`gf(x)`  can be broken down to a meaning as follows:

`f(x)`   means "do stuff" to `x`  (as we know already)

`g(x)`   means "do stuff" to `x` 

So

`gf(x)`   means do "`f`" stuff to `x` 

Then do "`g`" stuff to the outcome.

and similarly

`fg(x)`   means do "`g`" stuff to `x` 

Then do "`f`" stuff to the outcome.

Example 1

If `f(x)=x^2`  and `g(x)=x-3`

Find `fg(1)`

This is solved as follows:

`f(x)=x^2`   means the "stuff" you do to `x`  is square it.

`g(x)=x-3`   means the "stuff" you do to `x`  is subtract `3`.

`g(1)`   means do "stuff" to `x`  when `x=1`.

`fg(1)`   means do "stuff" to `x`  when `x=`  the result of `g(1)`.

So  `g(x)=x-3`

`g(1)=1-3`

`g(1)=-2`

and  `f(x)=x^2`

`f(g(1))=f(-2)=(-2)^2=4`

`fg(1)=4`

Example 2

If `f(x)=x^2`  and `g(x)=x-3`

Find `gf(1)`

This is solved as follows:

`f(x)=x^2`   means the "stuff" you do to `x`  is square it.

`g(x)=x-3`   means the "stuff" you do to `x`  is subtract `3`.

`f(1)`   means do "stuff" to `x`  when `x=1`.

`gf(1)`   means do "stuff" to `x`  when `x=`  the result of `f(1)`.

So   `f(x)=x^2`

`f(1)=1^2`

`f(1)=1`

and `g(x)=x-3`

`g(f(1))=g(1)-3=1-3=-2`

`gf(1)=-2`

Example 3

If `f(x)=x+2`  and `g(x)=3x`

Find `fg(5)`

This is solved as follows:

`f(x)=x+2`   means the "stuff" you do to `x`  is add `2`.

`g(x)=3x`   means the "stuff" you do to `x`  is multiply by `3`.

`g(5)`   means do "`g`  stuff" to `x`  when `x=5`.

`fg(5)`   means do "`f`  stuff" to `x`  when `x=`  the result of `g(5)`.

So   `g(x)=3x`

`g(5)=3times5`

`g(5)=15`

and `f(x)=x+2`

`f(g(5))=15+2`

`fg(5)=17`