Graph each rational function. ƒ(x)=(9x²-1)/(x²-4)

Verified step by step guidance

1. Identify the rational function given: \(f(x) = \frac{9x^2 - 1}{x^2 - 4}\).
2. Factor both the numerator and the denominator to simplify and find key features: factor \$9x^2 - 1\( as a difference of squares and \)x^2 - 4$ also as a difference of squares.
3. Determine the domain by finding values of \(x\) that make the denominator zero, since these values are excluded from the domain and correspond to vertical asymptotes or holes.
4. Find the vertical asymptotes by setting the denominator equal to zero and solving for \(x\), and check if any factors cancel with the numerator to identify holes instead.
5. Find the horizontal asymptote by comparing the degrees of the numerator and denominator polynomials, and analyze the end behavior of the function accordingly.

Key Concepts

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x). Understanding its domain, zeros, and behavior depends on analyzing both numerator and denominator polynomials. Key features include vertical asymptotes where the denominator is zero and holes where factors cancel.

Asymptotes of Rational Functions

Asymptotes are lines that the graph approaches but never touches. Vertical asymptotes occur at values making the denominator zero (if not canceled), while horizontal or oblique asymptotes describe end behavior based on the degrees of numerator and denominator polynomials.

Factoring and Simplifying Polynomials

Factoring polynomials helps identify zeros and simplify the function. By factoring numerator and denominator, common factors can be canceled, revealing holes in the graph and simplifying the analysis of intercepts and asymptotes.