Question

Express the function in the form f ∘ g. (use non-identity functions for f and g.) f(x) = 3 x 7 + 3 x

Answer 1

The composed function is
`3x^7+3x`. We are going to express this function as
`(f\circ g)(x)`,

where neither of the functions are identity functions, that is f(x)≠x, and g(x)≠x.


Let
`g(x)=3x`, and let
`\displaystyle{f(x)= ((1)/(3^6))x^7+x`


then


`f(g(x))=\displaystyle{ ((1)/(3^6))(g(x))^7+g(x)=\displaystyle{ ((1)/(3^6))(3x)^7+3x`


`=\displaystyle{ ((1)/(3^6))\cdot3^7x^7+3x=3x^7+3x`



Answer:
`g(x)=3x`,
`\displaystyle{f(x)= ((1)/(3^6))x^7+x`