Question

Let$f\left(x\right)=ax+b$ and $g\left(x\right)=cx+d$where a, b, c, and d are constants. Determine necessary and sufficient conditions on the constants a, b, c, and d so that$f\circ g=g\circ f$

Short answer

The function is$ad+b=bc+d$

Step-by-step solution

Step 1: 

Composition of f and g : $(f\circ g)\left(a\right)=f\left(g\right(a\left)\right)$

Step 2:

Given:

$\begin{array}{r}f\left(x\right)=ax+b\\ g\left(x\right)=cx+d\end{array}$

Use the definition of composition:

$\begin{array}{r}(f\circ g)\left(x\right)=f\left(g\right(x\left)\right)=f(cx+d)=a(cx+d)+b=acx+ad+b\\ (g\circ f)\left(x\right)=g\left(f\right(x\left)\right)=g(ax+b)=c(ax+b)+d=acx+bc+d\end{array}$

The two compositions have to be equal$(f\circ g=g\circ f)$

$acx+ad+b=acx+bc+d$

Subtract from each side of the previous equation:

$ad+b=bc+d$

Hence, the necessary and sufficient conditions on the constants is:$ad+b=bc+d$