We are given two functions:

• $f(x) = 1x - 3$
• $g(x) = 7x + 3$

Now, let's find the compositions of the functions.

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

To compute $f(g(x))$, we substitute $g(x) = 7x + 3$ into $f(x) = 1x - 3$: $f(g(x)) = f(7x + 3) = 1(7x + 3) - 3$ Simplify: $f(g(x)) = 7x + 3 - 3 = 7x$

Thus, $f(g(x)) = 7x$.

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

To compute $g(f(x))$, we substitute $f(x) = 1x - 3$ into $g(x) = 7x + 3$: $g(f(x)) = g(1x - 3) = 7(1x - 3) + 3$ Simplify: $g(f(x)) = 7x - 21 + 3 = 7x - 18$

Thus, $g(f(x)) = 7x - 18$.

Final Answers:

• $f(g(x)) = 7x$
• $g(f(x)) = 7x - 18$

Would you like more details or explanations?

Here are some related questions:

1. What is the inverse of $f(x)$ and $g(x)$?
2. Can you find $f(f(x))$ and $g(g(x))$?
3. What happens if you solve $f(x) = g(x)$?
4. What are the domain and range of $f(x)$ and $g(x)$?
5. How would you graph $f(x)$ and $g(x)$?

Tip: Function compositions are useful for combining multiple transformations in algebra.