Question

Let $f\left(x\right)=ax+b\text{and}g\left(x\right)=cx+d,a\ne 0,c\ne 0.$ Assume $a=1,b=2,$ If $\left(fog\right)\left(x\right)=\left(gof\right)\left(x\right)$ for all $x$ what can you say about c and d _________________.

• A. c and d both arbitrary
• B. $c=1,d$ arbitrary
• C. c arbitrary $d=1$
• D. $c=1,d=1$

Answer 1

Answer: (B)

$c=1,d$ arbitrary

$\begin{array}{cc}\text{Solution}-f\left(x\right)=& ax+bg\left(x\right)=cx+d\\ f\left(g\right(x\left)\right)& =a(cx+d)+b\\ & =acx+ad+b-\left(1\right)\\ g\left(f\right(x\left)\right)& =c(ax+b)+d\\ & =cax+cb+d-\left(2\right)\end{array}$

$\begin{array}{c}ACC\text{. to question}\left(1\right)=\left(2\right)\\ \Rightarrow acx+ad+b=acx+cb+d\\ \Rightarrow d+2=2c+d\\ c=1\\ \text{and}d=\text{arbitrary}\\ \text{Hemce,}\left(B\right)\text{is the correct option.}\end{array}$