Functions Question 464
Question: $ f(x)=| x-1 |,f:{R^{+}}\to R $ and $ g(x)=e^{x}, $ $ g:[(-1,\infty )\to R]. $ If the function fog (x) is defined, then its domain and range respectively are
Options:
A) $ (0,\infty )and[0,\infty ) $
B) $ [-1,\infty )and[0,\infty ) $
C) $ [-1,\infty )and[ 1-\frac{1}{e},\infty ) $
D) $ [-1,\infty )and[ \frac{1}{e}-1,\infty ) $
Show Answer
Answer:
Correct Answer: B
Solution:
[b] $ f(x)=| x-1 |= \begin{cases} 1-x,,0<x<1 \\ x-1,,x\ge 1 \\ \end{cases} . $ $ g(x)=e^{x},x\ge -1 $ $ (fog)(x)= \begin{cases} 1-g(x),,0<g(x)<1i.e.-1\le x<0 \\ g(x)-1,,g(x)\ge 1i.e.0\le x \\ \end{cases} . $ $ = \begin{cases} 1-e^{x},-1\le x<0 \\ e^{x}-1,x\ge 0 \\ \end{cases} . $
$ \therefore $ Domain $ =[-1,\infty ) $ fog is decreasing in $ [-1,0) $ and increasing in $ [0,\infty ) $ $ fog(-1)=1-\frac{1}{e} $ and $ fog(0)=0 $ As $ x\to \infty ,fog(x)\to \infty , $
$ \therefore $ range $ =[0,\infty ) $
BETA
