Decomposition of Functions

Algebra Level 2

$f(x)$ is a function composed from two other functions such that $f(x)=h(g(x))=(2x-1)^2$ .

Which of the following are possible functions for $h(x)$ and $g(x)\,$ ?

h(x)=x

and

g(x)=2x-1

g(x)=2x-1

and

h(x)=x^2

g(x)=4x-2

and

h(x)=2x^2

h(x)=2x-1

and

g(x)=x^3

1 solution

Michael B Staff
Jul 26, 2016

To avoid using guess and check we can look at the function to try to find how one function may be nested inside of another.

In this problem we see that $2x-1$ is nested inside of a squared function. The squared function could be represented as $x^2$ .

To finish decomposing our function we need to say that $h(x)=x^2$ and $g(x)=2x-1$ .