This write-up is for problems #3 in Assignment 1.

Linear Functions Tangent to their Product

by

The challenge of this exploration is to find two linear functions f(x) and g(x) such that their product

h(x) = f(x).g(x)

is tangent to each of f(x) and g(x) at two distinct points.

If f(x) = x and g(x) = -x+1 and h(x) = (x)(-x+1), then f(x) is tangent to h(x) at the origin and g(x) is tangent to h(x) at the point (1, 0 ).

The functions f(x) and g(x) are perpendicular lines. Since f(x) and g(x) have opposite slopes, their product will result in a parabola opening downward. We may see this more clearly if we expand h(x) as seen below.

If we explore the possibility of two functions f(x) and g(x) both with positive slopes having a product h(x) that is tangent to each of f(x) and g(x) at two distinct points, then it may be helpful to look at the graph below for f(x) = 4x + 1 and g(x) = x + 3 and h(x) = (4x + 1)(x + 3). Not only is h(x) not tangent to f(x) and g(x), but any relation tangent to f(x) and g(x) must not fail the vertical line test. This same argument would hold true for two functions f(x) and g(x) both having negative slopes.

If we conjecture that the slopes of f(x) and g(x) must be opposites, then we should explore the possibility of f(x) with a negative slope and g(x) with a positive slope. If f(x) = -x and g(x) = x + 1 and h(x) = (-x)(x + 1) then f(x) is tangent to h(x) at the origin and g(x) is tangent to h(x) at the point ( -1, 0 ).

If we let n equal some constant not equal to zero, then the two linear functions f(x) = nx +n and g(x) = -nx - (n - 1) have a product h(x) = (nx + n)(-nx - (n - 1)) that is tangent to f(x) and g(x) in two distinct points. These two tangent points are the roots of the function h(x). These tangent points are ( -1 , 0 ) and ( -(n-1)/n , 0 ). Furthermore, by using the opposite slopes in f(x) and g(x) we can arrive at two different functions f(x) = -nx + n and g(x) = nx - (n-1) that have a product h(x) = (-nx +n)(nx - (n-1)) that is tangent to f(x) and g(x) in two distinct points. Again these points are the roots of the function h(x). These tangent points are ( 1 , 0 ) and ( (n-1)/n , 0 ). The graph below occurs when n = 4.

As the absolute value of the slope increases for f(x) and g(x), these two linear functions approach the vertical line x = -1. Likewise, as the absolute value of the slope increases for f(x) and g(x), these two linear functions approach the vertical line x = 1. To view an animation of the graph directly above please click here. Please notice that when n = 0 we have only one horizontal line at y = 1.

Return