We are tasked with finding $f(g(x))$, where:

$f(x) = x^2 - 5x + 15$ $g(x) = -3x + 1$

To find $f(g(x))$, we substitute $g(x)$ into $f(x)$:

1. Start with $f(x) = x^2 - 5x + 15$, and replace every $x$ in this expression with $g(x) = -3x + 1$.

So, $f(g(x)) = ((-3x + 1)^2) - 5(-3x + 1) + 15$.

Now, let's expand and simplify this expression:

$f(g(x)) = (-3x + 1)^2 - 5(-3x + 1) + 15$

First, expand $(-3x + 1)^2$: $(-3x + 1)^2 = (9x^2 - 6x + 1)$

Next, expand $-5(-3x + 1)$: $-5(-3x + 1) = 15x - 5$

Now, substitute these expanded terms back into the expression: $f(g(x)) = (9x^2 - 6x + 1) + (15x - 5) + 15$

Combine like terms: $f(g(x)) = 9x^2 + (-6x + 15x) + (1 - 5 + 15)$ $f(g(x)) = 9x^2 + 9x + 11$

So, the final answer is: $f(g(x)) = 9x^2 + 9x + 11$

Would you like more details or have any questions?

Here are 5 related questions to deepen understanding:

1. How would you compute $g(f(x))$ for the same functions?
2. What is the degree of the polynomial resulting from $f(g(x))$?
3. What changes if $f(x)$ or $g(x)$ includes higher-degree terms?
4. How can you verify if $f(g(x))$ is a bijective function?
5. What is the significance of combining functions in this way in real-world applications?

Tip: When working with function composition, always pay close attention to substituting expressions accurately before expanding and simplifying.