The question in the image appears to be about finding the function composition of two functions. Here's the content from the image:

Use function composition to determine $f(g(x))$ and $g(f(x))$ for the functions:

$f(x) = 3x + 1 \quad \text{and} \quad g(x) = x^2 + 1$

Let's compute both $f(g(x))$ and $g(f(x))$.

1. Finding $f(g(x))$:

Given that $f(x) = 3x + 1$ and $g(x) = x^2 + 1$, to find $f(g(x))$, we substitute $g(x)$ into $f(x)$:

$f(g(x)) = f(x^2 + 1)$ Now substitute $x^2 + 1$ into $f(x) = 3x + 1$: $f(g(x)) = 3(x^2 + 1) + 1 = 3x^2 + 3 + 1 = 3x^2 + 4$

2. Finding $g(f(x))$:

To find $g(f(x))$, we substitute $f(x)$ into $g(x)$: $g(f(x)) = g(3x + 1)$ Now substitute $3x + 1$ into $g(x) = x^2 + 1$: $g(f(x)) = (3x + 1)^2 + 1$ First, expand $(3x + 1)^2$: $(3x + 1)^2 = 9x^2 + 6x + 1$ Now, add 1: $g(f(x)) = 9x^2 + 6x + 1 + 1 = 9x^2 + 6x + 2$

Final Results:

• $f(g(x)) = 3x^2 + 4$
• $g(f(x)) = 9x^2 + 6x + 2$

Would you like a deeper explanation or have any questions?

Here are some related questions:

1. What is the general concept of function composition in mathematics?
2. Can the composition of two functions be commutative, i.e., is $f(g(x))$ always equal to $g(f(x))$?
3. How does the degree of a polynomial change after composing two functions?
4. How would you compute $f(f(x))$ or $g(g(x))$ for these functions?
5. What happens to the function composition when one of the functions is a linear function?

Tip: Always expand expressions carefully when performing function composition to avoid algebraic errors.