The problem asks us to find $f(x)$ given that $f(x-2) = 3x - 5$.

Step 1: Substitute $x-2 = t$

Let $t = x - 2$. Then $x = t + 2$. Rewriting $f(x-2)$ in terms of $t$, we have: $f(t) = 3(t + 2) - 5$

Step 2: Simplify the expression

Substitute $t + 2$ into the equation: $f(t) = 3t + 6 - 5 = 3t + 1$

Step 3: Replace $t$ with $x$

Since $t = x$, we have: $f(x) = 3x + 1$

Final Answer:

The correct choice is: B) $f(x) = 3x + 1$

Would you like me to explain in more detail or cover related examples?

5 Related Questions:

1. How can you verify your solution for $f(x)$ using substitution?
2. What happens if the given function $f(x-2)$ had a different constant, e.g., $3x - 7$?
3. How do transformations like $f(x-2)$ affect the graph of $f(x)$?
4. Can this type of functional transformation be extended to quadratic or higher-degree functions?
5. What are the key differences between $f(x-2)$ and $f(x+2)$ in terms of behavior?

Tip:

When solving functional equations, always check if substituting values back into the original equation gives consistent results.