Mammoth Memory

Composite functions - several functions together

If you get `gf(x)`

Because functions can be denoted as an `f` or `g`  or `h`  or any letter.

`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`  (same as `f(x)`)

 

So

`gf(x)`   means "Do stuff" to `x` 

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

 

and similarly

`fg(x)`   means "Do 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))=-2^2`

`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))=1-3`

`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`

 

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