Electronic Proceedings of the Seventh Annual International Conference on Technology in Collegiate MathematicsOrlando, Florida, November 1720, 1994Paper C030Families of Linear Functions and Their Envelopes 
Steve LighDepartment of Mathematics Southeastern Louisiana University Hammond, LA 70402 USA Phone: (504) 5492175 LIGH@SELU.EDU list of all papers by this author  Randall G. WillsDepartment of Mathematics Southeastern Louisiana University Hammond, LA 70402 USA Phone: (504) 5492660 rwills@selu.edu list of all papers by this author 
Click to access this paper: 
Additional file(s) associated with this paper:

Consider the family of lines y=a*x+f(a), generated by a differentiable function y=f(x). We define the envelope of this set of lines to be a function y=g(x) such that every line in the family is tangent to the graph of y=g(x) at some point and through each point on the graph of y=g(x), there passes a line in the family that is tangent to y=g(x). We use the notation f(x)>g(x) to denote that the envelope of the family of lines y=a*x+f(a) is the function y=g(x). We present two main results:
Theorem 1 will describe the effect that translations and dilations of f(x) have on the envelope g(x), and Theorem 2 will give a closed form expression for the envelope g(x) for a certain class of functions f(x).
Theorem 1: Suppose f(x)>g(x), and let F(x)=A*f(B*x+C)+D*x+E, and G(x)=A*g(x/(A*B)+D/(A*B))(C/B)*x+((C*D)/B+E) where A, B, C, D, E are real numbers and A,B<>0. Then F(x)>G(x).
Theorem 2: Suppose f(x) is a differentiable function such that f'(x) is injective. f(x)>g(x) if and only if g(x)=f((f')^(1)(x))+x*(f')^(1)(x). Note: (f')^(1)(x) is the inverse of f'(x).