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, such as $f(x) = \frac{1}{x}$, has two branches and two asymptotes: the x-axis and y-axis.
2. Reciprocal functions have vertical asymptotes where the denominator equals zero.
3. The horizontal asymptote of a reciprocal function $\frac{1}{g(x)}$ is $y=0$ if the degree of $g(x)$ is greater than zero.
4. Reciprocal functions exhibit symmetry; for instance, $f(x) = \frac{1}{x}$ is symmetric about the origin.
5. Transformations such as translations, reflections, and dilations can be applied to reciprocal functions just like other functions.

Review Questions

• What are the vertical and horizontal asymptotes of the function $f(x) = \frac{1}{x-2}$?
• How does the graph of $f(x) = -\frac{1}{x}$ differ from that of $f(x) = \frac{1}{x}$?
• If given a polynomial in the denominator, how would you determine its vertical asymptotes?

Related terms

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.

Polynomial Function: A mathematical expression involving a sum of powers in one or more variables multiplied by coefficients.