☰ Contents · Algebra

Even and odd functions

Lessons 12 · 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
12

Even and odd functions

Textbook: pp. 46–50
GoalKnow the definitions of even and odd functions, decide whether a function is even or odd, and use the symmetry of the graph.
New words
even function · juft funksiyaodd function · toq funksiyasymmetry about the origin · koordinatalar boshiga nisbatan simmetriyasymmetry of the domain · aniqlanish sohasining simmetrikligi
Explanation

If y(−x) = y(x) for every x in the domain, the function is even; its graph is symmetric about the Oy axis (for example y = x², y = |x|, y = x⁴). If y(−x) = −y(x) for every x, the function is odd; its graph is symmetric about the origin (for example y = x³, y = 1 : x). A function can be neither even nor odd, for example y = 3x − 2. The domain of an even or odd function must be symmetric about 0: if x is in the domain, so is −x. That is why y = √x is neither even nor odd. To test, compute y(−x) and compare it with y(x) and −y(x). When graphing, draw the part for x ≥ 0 and then reflect it: an even function in the Oy axis, an odd function through the origin.

Worked examples
y = x⁴ − 3x²: y(−x) = (−x)⁴ − 3(−x)² = x⁴ − 3x² = y(x), so it is even. y = x³ + x: y(−x) = −x³ − x = −y(x), so it is odd. y = 3x − 2: y(1) = 1, y(−1) = −5; −5 ≠ 1 and −5 ≠ −1, so it is neither.
Graph of y = x³: for x ≥ 0 the points (0, 0), (1, 1), (2, 8). Since the function is odd, the points (−1, −1) and (−2, −8) also lie on it, symmetric about the origin.
Class activity

“Mirror and half-turn”: students mark several points for x ≥ 0 on squared paper; their partner copies them to x < 0 by the “even” or “odd” rule, and both check with the formula.

Practice
1
Which of y = x² + 5, y = 1 : x, y = x² + x are even, odd or neither?
2
For an odd function with y(2) = 7, what is y(−2)?
3
For an even function with y(3) = 5, what is y(−3)?
4
Why is y = √x neither even nor odd?