Rational Function Basics

Algebra 2 · Axiom Academy

LESSON Rational Function Basics A rational function is one polynomial divided by another, — and everything interesting happens where the denominator hits zero. 1. What Is a Rational Function? Just as a rational number is one integer over another, a rational function is one polynomial over another. Its value at each input is simply the numerator divided by the denominator. where P(x) and Q(x) are polynomials and Examples of rational functions The domain of a rational function is all real numbers except the inputs where the denominator equals zero — because at those inputs the function would ask us to divide by zero, which is undefined. Set the denominator equal to zero: Q(x) = 0 . Solve for x to find the restricted values. The domain is every real number except those values. Step 1. Set the denominator equal to zero: Domain: all real numbers except x = 3 . What happens near an input where the denominator is zero? The denominator shrinks toward zero, so the fraction grows without bound — the function values shoot off to positive or negative infinity. As x creeps closer to the restricted value, the denominator gets closer to zero, making the fraction larger and larger in absolute value. The two sides of a vertical asymptote can behave very differently. Let's approach x = 2 from each direction for and watch the values. x = 1.9 → f = 1/(−0.1) = −10 x = 1.99 → f = 1/(−0.01) = −100 x = 1.999 → f = 1/(−0.001) = −1000 → −∞

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