jmc

Note that $f(4)=5.$ If we set $k$ so that $x+4=5,$ then $k=1.$ This value of $k$ makes the function continuous, as shown below.



If $k>1,$ then the function no longer has an inverse since it fails the horizontal line test.



And if $k<1,$ then the function has an inverse, but that inverse is not defined for all real numbers. Specifically, the inverse is not defined on the interval $[k+4,5).$



Therefore, the only possible value of $k$ is $\boxed{1}.$