jmc

We take cases:

If $x<4,$ then $f(x)=(4-x)+(20-x)+(50-x)=74-3x.$

If $4\le x<20,$ then $f(x)=(x-4)+(20-x)+(50-x)=66-x.$

If $20\le x<50,$ then $f(x)=(x-4)+(x-20)+(50-x)=26+x.$

If $50\le x,$ then $f(x)=(x-4)+(x-20)+(x-50)=3x-74.$

These parts of the graph of $f(x)$ connect continuously; since the first two parts have negative slope while the last two parts have positive slope, it follows that the minimum value of $f(x)$ is attained at $x=20,$ which gives $f(x)=26+20=46.$ Therefore, the range of $f(x)$ is $\boxed{[46,\infty )}.$