Question

Given the following functions:

f(x)=x2
g(x)=x−3


Find the composition of the two functions and show your process:

f(g(x))

*** Caution: this is different from
g(f(x))

2. Given the following functions:

f(x)=x2
g(x)=x−3


Find the composition of the two functions and show your process:

g(f(x))


Caution: This is different from
f(g(x))

3. If the composition of two functions is:

1x−3


What would be the domain restriction? Describe how you found that answer.

Answer 1

1)
`{\bf f }\circ {\bf g}(x) ={\bf x^2}-{\bf6x}+{\bf9}` and
`{\bf g }\circ {\bf f}(x)={\bf x^2}-{\bf3}` are not equal

ie,
`{\bf f} \circ {\bf g}(x) \\eq {\bf g} \circ{\bf f}(x)`

2) The domain is x

Explanation:

Given functions are
`f(x)=x^2` and
`g(x) = x-3`

now find the composition of two functions verify that

`f \circ g = g {\circ} f`

now find the composition of
`f\circ g`

`f \circ g=f(g(x))`

`f \circ g=f(x-3)`

`f \circ g=(x-3)^2`

`f \circ g=x^2-2(x)(3)+3^2`

`f \circ g=x^2-6x+9`

now find the composition of
`g \circ f`

`g \circ f=g(f(x))`

`g \circ f=g(x^2)`

`g \circ f=x^2-3`

Comparing the above two compositions we get

`f \circ g = x^2-6x+9` and
`g \circ f= x^2-3` are not equal

ie,
`f \circ g \\eq g \circ f`.

2) Given that the composition of two function is x-3

Let the functons be f(x) and g(x)

so the composition of two function
`f \circ g=x-3`

it may be written as
`f(g(x))=x-3`

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

`g(x)=x-3`