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Topos Uranos
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Functions with radicals: domain, signs and graph

Summary

This lesson extends to functions with radicals the method of sign charts and of polynomial division: it determines their domain when there are roots in the numerator and in the denominator, studies their sign and their behavior at the endpoints of the domain, where the graph ends at a point or approaches a vertical asymptote, and finds their asymptotes for large values of the variable by taking the highest power of xx out of each root, paying attention to the cases of positive xx and negative xx. It ends with the range, which is determined exactly when the limits at the endpoints allow it, and with the complete analysis of functions whose radicand is a trinomial of degree 88.

Learning objectives

  1. Determine the domain of a function with radicals in the numerator and in the denominator by means of sign charts and the factorization of trinomials of degree 2n2n.
  2. Study the sign of a function with radicals and its behavior at the endpoints of the domain, distinguishing the endpoints at which the graph ends from the vertical asymptotes.
  3. Find the asymptotes of a function with radicals as xx tends to ∞\infty and to −∞-\infty, treating positive and negative values of the variable separately.
  4. Sketch the graph of a function with radicals and determine its range when the analysis allows it, recognizing the cases that require the techniques of differential calculus.
  5. Analyze functions with radicals that describe applied quantities, such as the difference of the distances to two antennas or the Lorentz factor of relativity, and interpret their domain, their asymptotes and their range in context.

The domain of a function with radicals

In the lesson on algebraic functions, the rules that fix the domain of a function defined by a formula were established: no denominator may equal zero, and the radicand of a root of even index may not be negative. When the root is in a denominator, both rules act at once, and its radicand must be strictly positive; when it is in the numerator, it suffices that it be nonnegative. Roots of odd index impose no condition at all, except that of not vanishing if they are in a denominator.

If the radicands are polynomials of degree greater than 11, each condition is a polynomial inequality, which is solved by factoring and building a sign chart, as in the lesson on sign charts and inequalities. The quadratic factors are obtained with the quadratic formula, and trinomials of the form ax2n+bxn+ca x^{2n} + b x^n + c are reduced to quadratics with the change of variable u=xnu = x^n, as in the lesson on factoring quadratic polynomials. A quadratic factor with negative discriminant always keeps the sign of its leading coefficient, so it contributes no critical points and can be removed from the chart.

The domain of a function with radicals is obtained by combining all the conditions at once: nonnegative radicand for each root of even index in the numerator, positive radicand for each root of even index in a denominator, and denominators different from zero. Each condition is solved with a sign chart, and the domain is the intersection of the sets obtained.