the function f is defined by f:x->x-3. another function g is such that gf:x->(x^2)+2. find function g

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.

The function f is defined by f:x->x-3. Another function g is such that gf:x->(x^2)+2. Find function g.

Explore how to determine the function g(x) when given f(x) = x - 3 and the composite function gf(x) = x^2 + 2. This problem involves understanding function composition and quadratic functions, suitable for high school students in Grades 10-12.

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