To express $f(x) = \sqrt{x^2 + 3}$ as the composition of two functions, we need to find two functions, say $g(x)$ and $h(x)$, such that:

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

1. Identify the inner function $h(x)$ and the outer function $g(x)$:

   • Notice that the expression inside the square root, $x^2 + 3$, could be represented as the inner function.
   • So, let: $h(x) = x^2 + 3$
   • Then, the square root operation is applied to $h(x)$, so let: $g(x) = \sqrt{x}$

2. Verify the composition:

   • Now, if we compose $g$ and $h$, we get: $g(h(x)) = g(x^2 + 3) = \sqrt{x^2 + 3}$
   • This matches $f(x)$.

Therefore, the functions $g(x)$ and $h(x)$ are: $g(x) = \sqrt{x} \quad \text{and} \quad h(x) = x^2 + 3$

Final Answer:

$f(x) = g(h(x)) \quad \text{where} \quad g(x) = \sqrt{x} \quad \text{and} \quad h(x) = x^2 + 3$

Would you like more details or have any questions?

Here are some related questions to deepen understanding:

1. What would the composition look like if $f(x) = \sqrt{x + 5}$?
2. How would you decompose $f(x) = (x^2 + 3)^2$ into two functions?
3. What is the purpose of decomposing functions in mathematical analysis?
4. Can you express $f(x) = \sqrt[3]{x + 4}$ as the composition of two functions?
5. How does the order of composition (i.e., $g(h(x))$ vs. $h(g(x))$) affect the result?

Tip:

When decomposing functions, look for natural "layers" or "steps" in the expression, like operations inside parentheses or functions within other functions.