## 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 +∞

- Limit of G(x) when x goes to a
- 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 -∞

- Limit of G(x) when x goes to a

## Example

F(x) and 1/F(x) graphs characteristics:

- Transformation of extrema: (4,2) minimum of F ⇒ (4, 0.5) maximum of 1/F
- Horizontal asymptote: Horizontal asymptote of F: y = 4 ⇒ horizontal asymptote of 1/F: y = 0.25
- Vertical asymptote: Vertical asymptote of F: x = 2 ⇒ x-intersect point of 1/F: (2. 0)
- X-intersect point: X-intersect point of F (1, 0) ⇒ vertical asymptote of 1/F: x = 1

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