## Graph of the Reciprocal of a Function

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### Graph of the Reciprocal of a Function

Reciprocal of a Function $\frac{1}{f(x)}$

The following guidelines are useful in order to sketch the reciprocal of a function given the graph of the original function:

Where $f(x)$ is positive or negative then $\frac{1}{f(x)}$ is also positive or negative respectively.

Where $f(x)$ has zero(s) then the reciprocal function $\frac{1}{f(x)}$ has vertical asymptote(s) and vice versa.

Where $f(x)$ has a horizontal asymptote at y=c then the reciprocal function $\frac{1}{f(x)}$ has also horizontal asymptote at $y=\frac{1}{c}$ .

Where the original function $f(x)$ is increasing then the reciprocal function $\frac{1}{f(x)}$ is decreasing.

Where the original function $f(x)$ is decreasing then the reciprocal function $\frac{1}{f(x)}$ is increasing.

If the original function $f(x)$ has a maximum at $(c,f(c))$ then the reciprocal function $\frac{1}{f(x)}$ has a minimum at $(c,\frac{1}{f(c)})$.

If the original function $f(x)$ has a minimum at $(c,f(c))$ then the reciprocal function $\frac{1}{f(x)}$ has a maximum at $(c,\frac{1}{f(c)})$.

If the original function $f(x)$ has a point of inflexion at $(c,f(c))$ then the reciprocal function $\frac{1}{f(x)}$ has also a point of inflexion at $(c,\frac{1}{f(c)})$.
elizabeth

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