Given Functions:
$f(x) = x^2 + 1$
$g(x) = 2x - 1$
$h(x) = x^2 - 4$
We need to find $h \circ (g \circ f)$ and $(h \circ g) \circ f$. Note that function composition is associative, meaning both expressions will yield the same result. Let's calculate them to confirm.
Finding $h \circ (g \circ f):$
Step 1: Find $(g \circ f)(x)$
First, substitute $f(x)$ into $g(x)$:
$$(g \circ f)(x) = g(f(x))$$
$$g(x^2 + 1) = 2(x^2 + 1) - 1$$
$$= 2x^2 + 2 - 1$$
$$= 2x^2 + 1$$
Step 2: Find $h((g \circ f)(x))$
Now, substitute the result from Step 1 into $h(x)$:
$$h(2x^2 + 1) = (2x^2 + 1)^2 - 4$$
$$= (4x^4 + 4x^2 + 1) - 4$$
$$= 4x^4 + 4x^2 - 3$$
Finding $(h \circ g) \circ f :$
Step 1: Find $(h \circ g)(x)$
First, substitute $g(x)$ into $h(x)$:
$$(h \circ g)(x) = h(g(x))$$
$$h(2x - 1) = (2x - 1)^2 - 4$$
$$= (4x^2 - 4x + 1) - 4$$
$$= 4x^2 - 4x - 3$$
Step 2: Find $((h \circ g) \circ f)(x)$
Now, substitute $f(x)$ into the result from Step 1:
$$((h \circ g) \circ f)(x) = 4(f(x))^2 - 4(f(x)) - 3$$
$$= 4(x^2 + 1)^2 - 4(x^2 + 1) - 3$$
$$= 4(x^4 + 2x^2 + 1) - 4x^2 - 4 - 3$$
$$= 4x^4 + 8x^2 + 4 - 4x^2 - 7$$
$$= 4x^4 + 4x^2 - 3$$
Conclusion :
Both compositions result in the same expression: $\boxed{4x^4 + 4x^2 - 3}$
The correct option is A.
