Given the functions $f: x \to x - 3$ and $gf: x \to x^2 + 2$, we need to find the function $g(x)$.

Let's denote $f(x) = x - 3$. The composition $g(f(x)) = x^2 + 2$.

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

Given that $g(f(x)) = x^2 + 2$, and knowing $f(x) = x - 3$, substitute $f(x)$ into the equation:

$g(f(x)) = g(x - 3)$

So, the equation becomes:

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

Step 2: Solve for $g(x)$

To find $g(x)$, replace $x - 3$ with some variable $u$ so that:

$u = x - 3$

Then, $x = u + 3$, and the equation becomes:

$g(u) = (u + 3)^2 + 2$

Step 3: Expand the expression

Now, expand $(u + 3)^2$:

$g(u) = (u + 3)^2 + 2 = u^2 + 6u + 9 + 2 = u^2 + 6u + 11$

Thus, the function $g(x)$ is:

$g(x) = x^2 + 6x + 11$

Final Answer:

The function $g(x)$ is $g(x) = x^2 + 6x + 11$.

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you verify that the function $g(x)$ is correct?
2. What is the composition of $f(g(x))$?
3. How does the function $g(x) = x^2 + 6x + 11$ behave graphically?
4. Can you find $g(x)$ if $gf:x \to x^2 - 2x + 1$?
5. How would the function $g(x)$ change if $f(x) = 2x - 3$?

Tip: When dealing with composite functions, it’s often helpful to rewrite the functions in simpler terms to see how they interact.