Question

question
decompose the following function. find g(x) and h(x) so that the when composed $g\left(h\left(x\
ight)\
ight)=f\left(x\
ight)={\left(5x - 2\
ight)}^{2}+3$
check the course notes for examples of function decomposition. additional examples can be found in section 3.4 of algebra and trigonometry: https://openstax.org/books/algebra-and-trigonometry-2e/pages/3-4-composition-of-functions
answer attempt 1 out of 20
g(x)= and h(x)=

Explanation:

Step1: Identify inner function

The inner function $h(x)$ is the expression inside the parentheses and squared term: $h(x) = 5x - 2$

Step2: Identify outer function

Replace $h(x)$ with $x$ in $f(x)$ to get $g(x)$: $g(x) = x^2 + 3$

Step3: Verify composition

Substitute $h(x)$ into $g(x)$: $g(h(x)) = (5x-2)^2 + 3 = f(x)$

Answer:

$g(x)=x^2 + 3$ and $h(x)=5x - 2$