How I can set up a transfer function of y = mx+c?
You can't. Your system is not a linear map. In general, a transfer function can only be derived from a system that's linear and time-invariant (LTI). The constant term violates this linearity.

Specifically, the requirements for a linear map are:

1) \$y(x_1 + x_2)=y(x_1)+y(x_2)\$ (additive)

2) \$y(a x)=a y(x)\$ (homogeneous)

If you plug-n-chug into both equations, the violation should be clear:

\$y(x)=mx+5\$

1)

\$y(x_1)=mx_1+5\$, \$y(x_2)=mx_2+5\$, \$y(x_1)+y(x_2)=mx_1+mx_2+10\$

\$y(x_1+x_2)=m(x_1+x_2)+5=mx_1+mx_2+5 \neq y(x_1)+y(x_2)\$

2)

\$y(ax)=max+5\$

\$ay(x)=a(mx+5)=max+5a\neq y(ax)\$