Analyze rational functions, find asymptotes, domain, range, intercepts, and graph behavior with step-by-step solutions. Perfect for algebra, calculus, and advanced mathematics.
Please enter valid polynomial expressions for numerator and denominator.
Comprehensive rational function analysis with detailed explanations
Find vertical, horizontal, and oblique asymptotes with detailed explanations
Determine domain and range with exclusion analysis and interval notation
Find x and y intercepts with precise calculations and coordinate points
Analyze function behavior as x approaches infinity and negative infinity
Understanding rational function analysis and graphing
Input the numerator and denominator of your rational function in polynomial form.
Choose from complete analysis, asymptotes only, domain/range, or intercepts.
Receive comprehensive analysis with asymptotes, domain, range, intercepts, and behavior.
x-values where denominator = 0
Compare degrees: n < d → y=0, n=d → y=a/b
x-values where numerator = 0
Common rational function patterns and their analysis
Vertical asymptote: x = 3, Horizontal asymptote: y = 1
Vertical asymptote: x = -1, Horizontal asymptote: y = 0
Vertical asymptotes: x = ±1, Horizontal asymptote: y = 1
Hole at (2, 0.25), Vertical asymptote: x = -2
Common questions about rational functions and graphing
A rational function is a function that can be expressed as the ratio of two polynomials, f(x) = P(x)/Q(x), where Q(x) ≠ 0.
Vertical asymptotes occur at x-values where the denominator equals zero (and the numerator doesn't equal zero at the same point).
Compare the degrees of numerator and denominator: if n < d, asymptote is y = 0; if n = d, asymptote is y = leading coefficient ratio; if n > d, no horizontal asymptote.
The domain is all real numbers except the x-values where the denominator equals zero (vertical asymptotes and holes).
X-intercepts occur where the numerator equals zero (and the denominator doesn't equal zero at the same point).
Holes occur when both the numerator and denominator have the same factor that can be canceled out, creating a removable discontinuity.
Oblique asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. Use polynomial long division to find the oblique asymptote.
End behavior depends on the degrees of numerator and denominator. As x approaches ±∞, the function approaches the horizontal or oblique asymptote.
Start by finding asymptotes, intercepts, and holes. Then plot key points and sketch the graph approaching asymptotes and following the end behavior.
Rational functions are used in physics (inverse square laws), economics (cost-benefit analysis), engineering (signal processing), and many other fields involving proportional relationships.
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