Question

Let \(f(x) = ax+b\) and \(g(x) = cx+d\). The condition \(f(g(x)) = g(f(x))\) holds for all \(x\) if and only if ...

• A. \(f(a) = f(c)\)
• B. \(f(b) = g(b)\)
• C. \(f(d) = g(b)\)
• D. \(f(c) = g(a)\)

Hint:

Compute both compositions and compare constants.

Answer 1

The Correct Option is C

Step 1: Equate at $x = 0$:
Put $x = 0$ in the identity: $f(g(0)) = f(d)$ and $g(f(0)) = g(b)$.

Step 2: Conclude:
So $f(d) = g(b)$ is necessary. It is also sufficient since the slopes $ac$ match automatically.

Final Answer:
The condition is $f(d) = g(b)$, option (C). \[ \boxed{f(d) = g(b)} \]