jika f(x) = x + 3 (gOf) (x) = X^2 + 6X + 7 ditanya g(x)

To find \(g(x)\), we need to use the given information about the functions \(f(x)\) and \((g \circ f)(x)\). 

### Given Information:

1. \(f(x) = x + 3\)
2. \((g \circ f)(x) = x^2 + 6x + 7\)

### Goal:
We want to determine the function \(g(x)\).

### Step-by-Step Solution:

The notation \((g \circ f)(x)\) means that we first apply \(f(x)\) and then apply \(g\) to the result of \(f(x)\). So,

\[
(g \circ f)(x) = g(f(x))
\]

Given \(f(x) = x + 3\), we substitute this into the expression for \((g \circ f)(x)\):

\[
(g \circ f)(x) = g(x + 3)
\]

We are also given that:

\[
(g \circ f)(x) = x^2 + 6x + 7
\]

So,

\[
g(x + 3) = x^2 + 6x + 7
\]

To find \(g(x)\), we need to rewrite the expression for \(g(x + 3)\) in terms of \(x + 3\). Let's set \(u = x + 3\), then \(x = u - 3\). Substitute \(x = u - 3\) into the expression:

\[
g(u) = (u - 3)^2 + 6(u - 3) + 7
\]

Now, expand and simplify this expression:

\[
g(u) = (u^2 - 6u + 9) + (6u - 18) + 7
\]

Combine like terms:

\[
g(u) = u^2 - 6u + 9 + 6u - 18 + 7
\]
\[
g(u) = u^2 - 2
\]

Since \(u = x + 3\), we can replace \(u\) back with \(x\):

\[
g(x) = x^2 - 2
\]

### Conclusion:
The function \(g(x)\) is:

\[
g(x) = x^2 - 2
\]

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. How would you find \(f(x)\) if you are given \(g(x)\) and \((g \circ f)(x)\)?
2. What is the composition \((f \circ g)(x)\) given \(f(x) = x + 3\) and \(g(x) = x^2 - 2\)?
3. How can you determine if a composite function is invertible?
4. What is the domain of the composite function \((g \circ f)(x)\) in this example?
5. How would you find \(h(x)\) if you know \((h \circ g)(x) = (g \circ h)(x)\)?

**Tip:** When solving for unknown functions in compositions, substitution is a powerful method for finding expressions step by step.

Explore how to find the function g(x) from the composition (g ∘ f)(x) = x^2 + 6x + 7 using step-by-step solutions. This problem covers composition of functions, quadratic equations, and function transformations. Suitable for high school and above.

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