Graphs of y = ⁿ√x and powers with a rational exponent
Compare the graphs of ³√x, ⁴√x and √x, find the domain of an nth root and of a power with a rational exponent, and compare numbers by monotony.
- Germany: grade 9
- Ukraine: grade 10
- Russia: grade 10
Guide
In Ukrainian Grade 10 (algebra and analysis, section 1) the arithmetic nth root of a is written ⁿ√a. The function y = ²ᵏ⁺¹√x is defined on the whole number line: its graph passes through the origin, the function is odd and increasing on ℝ, D(y) = ℝ, E(y) = ℝ.
The function y = ²ᵏ√x is defined on [0; +∞): D(y) = [0; + ∞), E(y) = [0; + ∞), it is increasing, neither even nor odd, and its graph starts at (0; 0).
The function y = xr, r ∈ ℚ, is a power function with a rational exponent. If the irreducible fraction mn is positive, the domain is [0; +∞); if it is negative, the domain is (0; +∞).
An increasing function is used to compare: for non-negative a and b, a < b gives ⁿ√a < ⁿ√b.