Question

If $f\left(x\right)=ax+b$, $g\left(x\right)=cx+d$, then $fg\left(x\right)=gf\left(x\right)$ is equivalent to:

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

Answer 1

The correct option is C

$f\left(d\right)=g\left(b\right)$

Explanation for the correct option.

Find the relation:

$\begin{array}{rcl}fg\left(x\right)& =& a\left(cx+d\right)+b\\ & =& acx+ad+b...\left(1\right)\end{array}$

Similarly,

$\begin{array}{rcl}gf\left(x\right)& =& c\left(ax+b\right)+d\\ & =& acx+cb+d...\left(2\right)\end{array}$

It is given that $fg\left(x\right)=gf\left(x\right)$, so

$\begin{array}{rcl}acx+ad+b& =& acx+cb+d\\ \Rightarrow ad+b& =& cb+d\\ \Rightarrow f\left(d\right)& =& g\left(b\right)\end{array}$

Hence, option C is correct.