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!