To find \((f \circ g)(x)\), also written as \((fog)(x)\), we need to substitute the function \(g(x)\) into the function \(f(x)\).

Given:
- \(f(x) = x^2 - 2x + 3\)
- \(g(x) = 5x + 2\)

We substitute \(g(x)\) into \(f(x)\):
\[
(f \circ g)(x) = f(g(x)) = f(5x + 2)
\]

Now, substitute \(5x + 2\) for \(x\) in \(f(x)\):
\[
f(5x + 2) = (5x + 2)^2 - 2(5x + 2) + 3
\]

Now, expand the terms:
\[
(5x + 2)^2 = 25x^2 + 20x + 4
\]
\[
-2(5x + 2) = -10x - 4
\]
So, the equation becomes:
\[
f(5x + 2) = 25x^2 + 20x + 4 - 10x - 4 + 3
\]

Now simplify:
\[
f(5x + 2) = 25x^2 + (20x - 10x) + (4 - 4 + 3)
\]
\[
f(5x + 2) = 25x^2 + 10x + 3
\]

Thus, \((f \circ g)(x) = 25x^2 + 10x + 3\).

Would you like any additional details or have any questions?

Here are five related questions:
1. What is \((g \circ f)(x)\)?
2. How do you find the inverse of \(f(x)\) and \(g(x)\)?
3. What is the domain of \((f \circ g)(x)\)?
4. How would you find \((f \circ g)(2)\)?
5. How do you determine if \(f(x)\) and \(g(x)\) are one-to-one functions?

**Tip**: When composing functions, always substitute the entire second function into the first, and simplify carefully!

Learn how to find the composition of two functions, (f ∘ g)(x), where f(x) = x² - 2x + 3 and g(x) = 5x + 2. This solution involves expanding and simplifying quadratic expressions. Perfect for students in Grades 9-12 studying algebra and function composition.

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