We are given two functions:

• $f(x) = x^2 + 2$
• $g(u) = \sqrt{2u - 2}$

We are asked to find $g(f(x))$, which means we need to evaluate the composition of the two functions $g$ and $f$. This can be done by substituting $f(x)$ into $g(u)$.

Step-by-step solution:

1. Substitute $f(x)$ into $g(u)$:

   • We know that $f(x) = x^2 + 2$, so we replace $u$ in the expression for $g(u)$ with $f(x)$: $g(f(x)) = g(x^2 + 2)$

2. Apply the definition of $g(u)$:

   • The function $g(u)$ is given by $g(u) = \sqrt{2u - 2}$, so we substitute $u = x^2 + 2$ into this expression: $g(x^2 + 2) = \sqrt{2(x^2 + 2) - 2}$

3. Simplify the expression inside the square root:

   • First, distribute the 2 inside the parentheses: $g(x^2 + 2) = \sqrt{2x^2 + 4 - 2}$
   • Simplify the constants: $g(x^2 + 2) = \sqrt{2x^2 + 2}$

Thus, the composition $g(f(x))$ is: $g(f(x)) = \sqrt{2x^2 + 2}$

Final answer:

$g(f(x)) = \sqrt{2x^2 + 2}$

Would you like more details on this solution or have any further questions?

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Here are some related questions for further exploration:

1. How would the composition $f(g(x))$ differ from $g(f(x))$?
2. What would happen if we had $f(x) = x^2 - 2$ instead of $f(x) = x^2 + 2$?
3. Can you simplify $\sqrt{2x^2 + 2}$ further, for example, by factoring out a constant?
4. How can you visualize the function $g(f(x))$ on a graph?
5. What is the domain of the function $g(f(x))$?

Tip: When working with function compositions, always ensure you correctly substitute and simplify the expression step by step to avoid mistakes!