Question: Express your answer as a polynomial in standard form.
f, of, x, equals, 3, x, squared, plus, 5, x, plus, 13
f(x)=
3x^2 + 5x + 13
g, of, x, equals, x, minus, 3
g(x)= x - 3
Find: (f\circ g)(x)
1. **State the problem:** We are given two functions:
$$f(x) = 3x^2 + 5x + 13$$
$$g(x) = x - 3$$
We need to find the composition \((f \circ g)(x) = f(g(x))\).
2. **Recall the composition formula:**
$$(f \circ g)(x) = f(g(x))$$
This means we substitute $g(x)$ into every $x$ in $f(x)$.
3. **Substitute $g(x)$ into $f(x)$:**
$$f(g(x)) = 3(g(x))^2 + 5(g(x)) + 13$$
Substitute $g(x) = x - 3$:
$$= 3(x - 3)^2 + 5(x - 3) + 13$$
4. **Expand the squared term:**
$$(x - 3)^2 = (x - 3)(x - 3) = x^2 - 3x - 3x + 9 = x^2 - 6x + 9$$
So:
$$f(g(x)) = 3(x^2 - 6x + 9) + 5(x - 3) + 13$$
5. **Distribute the coefficients:**
$$= 3x^2 - 18x + 27 + 5x - 15 + 13$$
6. **Combine like terms:**
$$3x^2 + (-18x + 5x) + (27 - 15 + 13) = 3x^2 - 13x + 25$$
7. **Final answer:**
$$(f \circ g)(x) = 3x^2 - 13x + 25$$