 # 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