To solve $f(g(x))$ using the provided equations:

1. $f(x) = 2x + 1$
2. $g(x) = x - 3$

Step 1: Substitute $g(x)$ into $f(x)$

We need to find $f(g(x))$, which means we substitute $g(x)$ into $f(x)$. Specifically: $f(g(x)) = f(x - 3)$

Step 2: Use the definition of $f(x)$

The function $f(x) = 2x + 1$. Replace $x$ in $f(x)$ with $g(x) = x - 3$: $f(x - 3) = 2(x - 3) + 1$

Step 3: Simplify the expression

Expand and simplify: $f(x - 3) = 2x - 6 + 1$ $f(x - 3) = 2x - 5$

Final Answer:

$f(g(x)) = 2x - 5$

Thus, the boxed values are:

• $[2]$ for the coefficient of $x$,
• $[-5]$ for the constant.

Would you like further clarification or a breakdown of any step?

5 Related Questions:

1. What does the composition of functions represent in mathematics?
2. How would $g(f(x))$ be calculated using the same functions $f(x) = 2x + 1$ and $g(x) = x - 3$?
3. How can we verify the result of a composition of functions?
4. What is the general rule for composing two linear functions?
5. How does the domain of $f(g(x))$ differ from the domains of $f(x)$ and $g(x)$?

Tip:

When solving compositions, always substitute carefully and simplify step by step to avoid errors.