Question

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

• 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 $f\left(d\right)=g\left(b\right)$

$f\left(g\right(x\left)\right)=a(cx+d)+b$

$=acx+ad+b$...(i)

$g\left(f\right(x\left)\right)=c(ax+b)+d$

$=ac\left(x\right)+bc+d$ ...(ii)

$f\left(g\right(x\left)\right)=g\left(f\right(x\left)\right)$

From (i) and (ii)

$ac\left(x\right)+bc+d=acx+ad+b$

$cb+d=ad+b$

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