Functions Question 446
Question: Let $ f(x)=ax+b $ and $ g(x)=cx+d,\ a\ne 0,\ c\ne 0 $ . Assume $ a=1,\ b=2 $ . If $ (fog)(x)=(gof)(x) $ for all x, what can you say about c and d
[AMU 2000]
Options:
A) c and d both arbitrary
B) $ c=1,\ d $ arbitrary
C) c arbitrary, $ d=1 $
D) $ c=1,\ d=1 $
Show Answer
Answer:
Correct Answer: B
Solution:
$ (fog)(x)=f(g(x))=a(cx+d)+b $ and $ (gof)(x)=g(f(x))=c(ax+b)+d $ Given that, $ (fog)(x)=(gof)(x) $ and at $ a=1,,b=2 $
Þ $ \underset{x\to 0}{\mathop{\lim }},3,\frac{\tan 3x}{3x}+\underset{x\to 0}{\mathop{\lim }},\cos x=3+1=4 $
Þ $ c=1 $ and d is arbitrary.
BETA
