Question

decomposing composite functions
two functions represent the composite function $h(x) = (x - 1)^3 + 10$ so that $h(x) = (g \circ f)(x)$. given $f(x) = x + a$ and $g(x) = x^3 + b$, what values of $a$ and $b$ would make the composition true?
$a = \square$
$b = \square$

Explanation:

Step1: Define composite function

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

Step2: Substitute $f(x)$ into $g$

$g(f(x)) = (x+a)^3 + b$

Step3: Match to $h(x)$

Set $(x+a)^3 + b = (x-1)^3 + 10$

Step4: Solve for $a$

Compare linear terms: $a = -1$

Step5: Solve for $b$

Compare constants: $b = 10$

Answer:

$a = -1$
$b = 10$