Given the functions $f(x) = 2x + 3$, $g(x) = x^2 - 4$, and $h(x) = x - 2$, we are tasked with determining $(h \circ g \circ f)(x)$.

This means we need to compute the composition of these functions step by step. Here's how to approach it:

Step 1: Compute $g(f(x))$

We start by finding $g(f(x))$, which means substituting $f(x)$ into $g(x)$:

$f(x) = 2x + 3$

Substitute into $g(x)$:

$g(f(x)) = (2x + 3)^2 - 4$

Now expand $(2x + 3)^2$:

$(2x + 3)^2 = 4x^2 + 12x + 9$

So,

$g(f(x)) = 4x^2 + 12x + 9 - 4 = 4x^2 + 12x + 5$

Step 2: Compute $h(g(f(x)))$

Now, substitute $g(f(x)) = 4x^2 + 12x + 5$ into $h(x)$:

$h(g(f(x))) = (4x^2 + 12x + 5) - 2$

Simplify:

$h(g(f(x))) = 4x^2 + 12x + 3$

Final Answer:

Thus, $(h \circ g \circ f)(x) = 4x^2 + 12x + 3$.

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Do you have any questions or would you like further details on any of the steps?

Here are 5 related questions to deepen your understanding:

1. What is the general formula for composing two functions?
2. How do you expand a binomial like $(2x + 3)^2$?
3. Can you apply the composition of functions to more than three functions?
4. How would the result change if $f(x) = x - 1$ instead of $f(x) = 2x + 3$?
5. What is the inverse of $h(x) = x - 2$?

Tip: When composing functions, always start by working from the innermost function to the outermost function.