Reciprocal function

Definition

A reciprocal function is a function of the form $f(x) = \frac{1}{g(x)}$, where $g(x)$ is a non-zero polynomial. The simplest example is $f(x) = \frac{1}{x}$.

5 Must Know Facts For Your Next Test

1. The graph of a basic reciprocal function $f(x) = \frac{1}{x}$ has two asymptotes: the x-axis (horizontal asymptote) and the y-axis (vertical asymptote).
2. Reciprocal functions can have vertical asymptotes at values of $x$ where $g(x) = 0$.
3. The domain of a reciprocal function excludes values that make the denominator zero.
4. The range of most simple reciprocal functions like $f(x) = \frac{1}{x}$ is all real numbers except zero.
5. Transformations such as translations, reflections, stretches, and compressions can be applied to the graphs of reciprocal functions.

Review Questions

• What are the asymptotes for the function $f(x) = \frac{1}{x-3}$?
• If given a polynomial in the denominator, how do you determine the vertical asymptotes?
• How does adding a constant to the numerator or denominator affect the graph of a reciprocal function?

Related terms

Rational Function:

A rational function is any function that can be expressed as the ratio of two polynomials.

Asymptote:

A line that a graph approaches but never touches or crosses.

Transformation: Operations that alter the form of a function's graph, including translations, reflections, stretches, and compressions.