To find $(f \circ g)(x)$, also written as $(fog)(x)$, we substitute $g(x)$ into $f(x)$.

Given:

• $f(x) = x + 6$
• $g(x) = 3x^7 + 6x + 3$

Step 1: Substitute $g(x)$ into $f(x)$

We replace $x$ in $f(x)$ with $g(x)$: $(f \circ g)(x) = f(g(x)) = g(x) + 6$

Step 2: Simplify $f(g(x))$

Substitute $g(x) = 3x^7 + 6x + 3$ into $f(x)$: $f(g(x)) = (3x^7 + 6x + 3) + 6$

Combine like terms: $f(g(x)) = 3x^7 + 6x + 9$

Final Answer:

$(fog)(x) = 3x^7 + 6x + 9$

Would you like further explanation or have other questions?
Here are 5 related questions to deepen understanding:

1. What is the definition of function composition?
2. How do you simplify polynomials in function composition?
3. What happens if $f(x)$ or $g(x)$ has more complex terms?
4. Can we reverse the order and compute $(gof)(x)$?
5. How does composition affect the degree of a polynomial?

Tip: Always double-check substitutions step by step to avoid mistakes in combining terms.