Rational Function Analyzer

Intro: Given f(x) = P(x)/Q(x), finds domain, intercepts, vertical asymptotes, holes, and end behavior (horizontal/oblique asymptotes).

Worked example

• Find domain, vertical asymptotes, and holes for $f(x) = \dfrac{x^2 − 1}{x^2 − x − 2}$.



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• We have $f(x)=\dfrac{x^2-1}{x^2-x-2}$. First factor numerator and denominator.



• Factor the numerator: $x^2-1=(x-1)(x+1)$.



• Factor the denominator: $x^2-x-2=(x-2)(x+1)$.



• Rewrite the function: $f(x)=\dfrac{(x-1)(x+1)}{(x-2)(x+1)}$.



• We see a common factor $(x+1)$ in numerator and denominator. This indicates a potential hole at $x=-1$ (if the factor cancels).



• Cancel the common factor to obtain the simplified form $f(x)=\dfrac{x-1}{x-2}$, but remember that $x\neq -1$ from the original domain.



• The domain excludes values where the original denominator is zero: $x^2-x-2=0 \Rightarrow x=-1$ or $x=2$. So $x \neq -1,2$.



• Because $(x+1)$ canceled, $x=-1$ is a hole, not a vertical asymptote. The hole is at $x=-1$ with y-value given by the simplified function: $f(-1) = (-1-1)/(-1-2) = -2/-3 = 2/3$.



• The remaining factor $x-2$ in the denominator gives a vertical asymptote at $x=2$.



• Answer summary: domain is all real x except $x=-1,2$; hole at $(-1, 2/3)$; vertical asymptote at $x=2$.

Why choose MathGPT?

• Get clear, step-by-step solutions that explain the “why,” not just the answer.
• See the rules used at each step (power, product, quotient, chain, and more).
• Optional animated walk-throughs to make tricky ideas click faster.
• Clean LaTeX rendering for notes, homework, and study guides.

How this calculator works

• Type or paste your function (LaTeX like \sin, \ln works too).
• Press Generate a practice question button to generate the derivative and the full reasoning.
• Review each step to understand which rule was applied and why.
• Practice with similar problems to lock in the method.

Related

• Asymptote Finder
• Domain & Range