Lessons 11 · 1 lessons · Sh.A. Alimov, A.R. Xalmukhamedov, M.A. Mirzaakhmedov. Algebra: textbook for Grade 9 of general secondary schools. Revised 4th edition. Tashkent: “O‘qituvchi”, 2019
11
Increasing and decreasing functions
Textbook: pp. 41–45
GoalKnow when the power function y = xʳ increases or decreases; prove increase and decrease from the definition.
New words
power function · darajali funksiyaincreasing function · o‘suvchi funksiyadecreasing function · kamayuvchi funksiyaexponent · daraja ko‘rsatkichi
Explanation
A function of the form y = xʳ is called a power function (r is a given number). For x > 0 it is defined for any r. If for any x₁ > x₂ in an interval we have y(x₁) > y(x₂), the function is increasing on that interval; if y(x₁) < y(x₂), it is decreasing. The key rule: if r > 0, then y = xʳ increases on x > 0; if r < 0, it decreases on x > 0. This is proved by raising the inequality x₁ > x₂ > 0 to the power r. So the equation xʳ = b (b > 0) has exactly one root x = b^(1/r) for x > 0. To prove that some other function increases we take x₁ > x₂ and check the sign of the difference y(x₁) − y(x₂).
Worked examples
Compare 7⁻² and 5⁻²: r = −2 < 0, the function decreases for x > 0, and 7 > 5, so 7⁻² < 5⁻². Check: 1/49 < 1/25.
Solve x^(2/3) = 4 (x > 0). x = 4^(3/2) = (√4)³ = 8. There is no other root, because for r = 2/3 > 0 the function increases and takes each value once. Check: 8^(2/3) = (∛8)² = 4.
Class activity
“Up or down?”: the teacher names functions y = xʳ (r = 3, −1, 1/2, −2/3...). Students raise a hand up (increasing) or down (decreasing) and must give the reason.
Practice
1
Do y = x⁻³, y = x^(1/3), y = x⁷ increase or decrease for x > 0?