Let's solve this step by step.

We are given the two functions:

• $f(x) = x^2 + 5$
• $g(x) = x + 3$

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

To find $f(g(x))$, we need to substitute $g(x)$ into the function $f(x)$. This means wherever there is an $x$ in $f(x)$, we replace it with $g(x)$.

$f(g(x)) = f(x + 3)$ Now, substitute $x + 3$ into the formula for $f(x)$: $f(x + 3) = (x + 3)^2 + 5$ Expand $(x + 3)^2$: $f(x + 3) = (x^2 + 6x + 9) + 5$ Simplify the expression: $f(g(x)) = x^2 + 6x + 14$

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

To find $g(f(x))$, we need to substitute $f(x)$ into the function $g(x)$. This means wherever there is an $x$ in $g(x)$, we replace it with $f(x)$.

$g(f(x)) = g(x^2 + 5)$ Now, substitute $x^2 + 5$ into the formula for $g(x)$: $g(x^2 + 5) = (x^2 + 5) + 3$ Simplify the expression: $g(f(x)) = x^2 + 8$

Final Answers:

• $f(g(x)) = x^2 + 6x + 14$
• $g(f(x)) = x^2 + 8$

Would you like further details or have any specific questions about this process?

Here are 5 related questions:

1. How would the result change if the functions were different, such as $f(x) = 2x + 1$ and $g(x) = x - 4$?
2. How do you perform function composition when dealing with more complex functions, such as trigonometric functions?
3. What are the general rules for composing functions?
4. How do you check if two functions are inverses of each other using composition?
5. Can you identify situations when function composition is undefined?

Tip: Always carefully expand expressions when composing functions, as small mistakes in algebra can lead to incorrect answers.