How to plot the graph of the reciprocal of a function ?

 

Given the function y = F(x), the question is how to define and plot the graph of G(x) = 1/ F(x).

A number of simple rules listed below describe how the graph characteristics of F (extrema, asymptotes, X-intercept points) are transformed in the graph of 1/F.

In addition, the following applies:

• Point-by-point correspondence: if (x,y) is a point of F(x), then (x, 1/y) is the corresponding point of G(x),
• If F(x) is increasing in a given interval of x, then G(x) = 1/F(x) is decreasing in the same interval,
• Vice-versa: if F(x) is decreasing in a given interval of x, then G(x) = 1/F(x) is increasing in the same interval.

 

Rule 1: Transformation of extrema

• If (a,b) is a maximum of F(x) , then (a,1/b) is a minimum of G(x),
• If (a,b) is a minimum of F(x) , then (a,1/b) is a maximum of G(x).

 

Rule 2: Horizontal asymptotes

A horizontal asymptote of F(x) e.g. y = b becomes a horizontal asymptote y = 1/b  of G(x).

In other words:

• If Limit of F(x) when x goes to -∞ is b, then:  Limit of G(x) when x goes to -∞   is 1/b,
• If Limit of F(x) when x goes to +∞ is b, then:  Limit of G(x) when x goes to +∞   is 1/b .

 

Rule 3: Vertical asymptotes

A vertical asymptote x = a  of F(x) becomes a x-intersect point (a, 0) of G(x).

In other words: If Limit of F(x) when x goes to a is +∞ or -∞, then:  G(a) = 0 .

 

Rule 4: X-intersect points - this is the reciprocal of rule 3 -

A x-intersect point of F(x) for instance a such that F(a) = 0 becomes a vertical asymptote x = a of G(x).

In other words:

• If F(a) = 0 and F is increasing at point a, then:
  • Limit of G(x) when x goes to a^- is -∞
  • Limit of G(x) when x goes to a⁺ is +∞
• If F(a) = 0 and F is decreasing at point a, then:
  • Limit of G(x) when x goes to a^- is +∞
  • Limit of G(x) when x goes to a⁺ is -∞

Example