To find $f(g(x))$, we will substitute the expression for $g(x)$ into the function $f(x)$.

Given:

• $f(x) = 2x - 7$
• $g(x) = 3x^2 - 5x - 7$

Now, to find $f(g(x))$, substitute $g(x)$ into $f(x)$:

$f(g(x)) = 2(3x^2 - 5x - 7) - 7$

Now, distribute the $2$ across the terms inside the parentheses:

$f(g(x)) = 2(3x^2) + 2(-5x) + 2(-7) - 7$

This simplifies to:

$f(g(x)) = 6x^2 - 10x - 14 - 7$

Finally, combine like terms:

$f(g(x)) = 6x^2 - 10x - 21$

So, the polynomial in standard form is:

$f(g(x)) = 6x^2 - 10x - 21$

Would you like further clarification on any part of this solution?

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Here are 5 related questions you can explore:

1. What is $g(f(x))$ for the same functions $f(x)$ and $g(x)$?
2. What is $f(f(x))$ for $f(x) = 2x - 7$?
3. How would you find the inverse of $f(x) = 2x - 7$?
4. What is the derivative of $f(g(x))$?
5. Can you find the x-intercepts of $f(g(x))$?

Tip: When performing composition of functions, always remember to substitute one function fully into the other, following the order indicated.