Rational functions are mathematical expressions defined as the quotient of two polynomials. This text delves into their simplification, identification of asymptotes, graphing methods, and the process of determining their inverses. Simplification involves factoring and reducing common factors, while graphing requires careful consideration of asymptotes and intercepts. Understanding the inverse of these functions is also crucial for comprehensive insights into their behavior.
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Rational functions are functions represented as the quotient of two polynomials, with the stipulation that the denominator cannot equal zero
The standard form of a rational function is \( f(x) = \frac{p(x)}{q(x)} \), where the denominator must include at least one variable term
Rational functions can be simplified by factoring the numerator and denominator and canceling out common factors
Rational functions can have vertical, horizontal, or oblique asymptotes, which describe the function's behavior near certain critical points
Vertical asymptotes occur at values of \( x \) that make the denominator zero, while horizontal asymptotes are determined by comparing the degrees of the numerator and denominator
Asymptotes are depicted as dashed lines on a graph, and the function's behavior near them must be taken into account when plotting points
Graphing rational functions involves identifying asymptotes, finding intercepts, and plotting additional points to create a sketch of the function
The graph of a rational function may consist of multiple disjointed curves, reflecting the function's domain and range
When graphing rational functions, it is important to carefully plot points around asymptotes and intercepts to accurately represent the function
The inverse of a rational function is found by interchanging the input and output of the original function and expressing it in terms of the original input
To compute the inverse, one replaces \( f(x) \) with \( y \), exchanges \( x \) and \( y \) in the equation, and isolates \( y \) to express it in terms of \( x \)
The inverse of a rational function can be verified by confirming that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \) for all values in the domain of the inverse function
00
Vertical Asymptote Identification
Occurs where denominator equals zero; function undefined.
01
Horizontal Asymptote Rules
If numerator's degree is less, asymptote is x-axis; if degrees equal, it's the leading coefficients' ratio.
02
Oblique Asymptote Condition
Exists when numerator's degree is one more than denominator's; found using long division.
