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 $f(x) = \frac{1}{x}$.
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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).
Reciprocal functions can have vertical asymptotes at values of $x$ where $g(x) = 0$.
The domain of a reciprocal function excludes values that make the denominator zero.
The range of most simple reciprocal functions like $f(x) = \frac{1}{x}$ is all real numbers except zero.
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.