Let's work through each example one by one.

Example 1

We are given:

• $f(x) = 3x$
• $g(x) = x - 4$

The problem asks for $g(f(3))$.

1. Find $f(3)$: $f(3) = 3 \times 3 = 9$

2. Substitute $f(3)$ into $g(x)$: $g(f(3)) = g(9) = 9 - 4 = 5$

So, the answer for Example 1 is: $g(f(3)) = 5$

Example 2

For this example, we have:

• $f(x) = 3x - 4$
• $g(x) = 2f(x) + 3$

We are asked to find the value of $k$ such that $g(k) = 0$.

1. Express $g(x)$ in terms of $f(x)$: $g(x) = 2f(x) + 3$

2. Substitute $f(x) = 3x - 4$ into $g(x)$: $g(x) = 2(3x - 4) + 3$ Simplify this: $g(x) = 6x - 8 + 3 = 6x - 5$

3. Set $g(k) = 0$ and solve for $k$: $6k - 5 = 0$ $6k = 5$ $k = \frac{5}{6}$

So, the answer for Example 2 is: $k = \frac{5}{6}$

Would you like more details on any step or have additional questions?

Follow-Up Questions

1. What is the general process of composing two functions like in Example 1?
2. How would we handle the composition if one function is quadratic, such as $f(x) = x^2$?
3. Could we reverse Example 1 and find $f(g(x))$ instead?
4. What if $g(x) = 0$ for Example 2? How would we solve for $x$?
5. How does the distributive property help in simplifying expressions like $2f(x) + 3$ in Example 2?

Tip

When dealing with compositions, always work from the innermost function outward to simplify step by step.