If f(x)=ax-b and g(x)=cx+d are such that f(g(x))=g(f(x)), then which one of the following holds ?
- A. f(d)=g(b)
- B. f(b)+g(d)=0
- C. f(a)+g(c)=2a
- D. f(d)+g(b)=2d ✓
Correct Answer: D. f(d)+g(b)=2d
Explanation
f(g(x)) = a(cx+d)-b = acx+ad-b. g(f(x)) = c(ax-b)+d = acx-bc+d. Equating them gives ad-b = d-bc \implies ad+bc = b+d. We evaluate f(d)+g(b) = (ad-b)+(bc+d) = ad+bc+d-b. Substituting ad+bc = b+d, we get (b+d)+d-b = 2d.
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