Question

If $f\left(x\right)=ax+b$ and $g\left(x\right)=cx+d$, then $f\left(g\right(x\left)\right)=g\left(f\right(x\left)\right)\iff$

• 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. $f\left(x\right)=g\left(a\right)$

Answer 1

Answer: (C)

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

We have $f\left(x\right)=ax+b,g\left(x\right)=cx+d$
Therefore, $f\left\{g\right(x\left)\right\}=g\left\{f\right(x\left)\right\}\iff f(cx+d)=g(ax+b)$
$\iff a(cx+d)+b=c(ax+b)+d$
$\iff ad+b=cb+d\iff f\left(d\right)=g\left(b\right)$