School curriculum· 30 min· 21 questions
Learn first: Hyperbola, root and absolute-value functions
You can learn a lot about a function’s “behaviour” without drawing its graph: is it symmetric, where does it increase, where is it positive, does it repeat, within what limits does it vary. These properties are asked about often in tests and greatly simplify hard problems: parity halves the work, monotonicity guarantees a unique root, periodicity reduces a large argument to a small one. This lesson continues “The concept of a function” and prepares for investigating functions with derivatives.
First of all, the domain must be symmetric about `0`: if x is in the domain, so is −x. Otherwise the function is neither even nor odd.
Even: f(−x) = f(x) — the graph is symmetric about the Oy axis
Odd: f(−x) = −f(x) — the graph is symmetric about the origin
Examples: even — x², x⁴, |x|, cos x; odd — x, x³, sin x, tg x, ctg x, 1/x. The names come from power functions: xⁿ is even for even n and odd for odd n.
Rules for operations (like signs when multiplying numbers):
Checking procedure: is the domain symmetric → compute f(−x) and compare it with f(x) and −f(x).
Determine the parity of f(x) = sin x · √(1 + x²).
Another example: g(x) = √x + x⁴ has domain x ≥ 0, which is not symmetric, so it is neither even nor odd.
Increasing: x₁ < x₂ ⇒ f(x₁) < f(x₂); decreasing: x₁ < x₂ ⇒ f(x₁) > f(x₂)
Monotonicity is checked on an interval. For example, y = x² decreases on (−∞; 0] and increases on [0; +∞), but is not monotonic on the whole line. y = 1/x decreases on (−∞; 0) and on (0; +∞) separately, but not on their union: −1 < 1, yet f(−1) = −1 < f(1) = 1.
Using monotonicity. An increasing function takes each value at most once. So in 2ˣ + x = 3 the left side increases and the root is unique — by inspection x = 1. Likewise, for an increasing function the inequality f(a) < f(b) is equivalent to a < b.
f(x + T) = f(x) for all x; the smallest positive T is the fundamental period
For a periodic function a large argument can be reduced by whole periods: if the period is T, then f(x + nT) = f(x).
Find the period of y = sin(x/3) + tg(x/2).
f is an odd function with period 4 and f(1) = 2. Find f(7).
A function is bounded above if there is M with f(x) ≤ M, and bounded below if f(x) ≥ m. If both hold, it is bounded. For example, sin x is bounded, x² only below, x³ not at all.
The range is E(f). For y = sin x, E = [−1; 1].
For y = a sin x + b cos x: E = [−√(a² + b²); √(a² + b²)]
This follows from introducing an auxiliary angle: a sin x + b cos x = √(a² + b²)·sin(x + φ).
Find the greatest value of y = 3 sin x + 4 cos x − 2.
The zeros of a function are the values of x with f(x) = 0; there the graph crosses the Ox axis. Zeros and points where the function is undefined split the domain into intervals of constant sign: inside each, a continuous function does not change sign. This is the basis of the interval method.
Where is y = (x − 1)(x + 2)/(x − 3) positive?