Parity of a function
A prediction algorithm measures its error symmetrically: being wrong by overestimating or underestimating by the same amount counts the same. This mathematical symmetry, already seen in ESO for simple polynomials, is generalised in Bachillerato to rational, trigonometric and composite functions.
Revision: checking parity by substituting $-x$
Determine analytically whether the polynomial function $f(x) = 3x^4 - 2x^2 + 5$ is even, odd, or neither.
Determine analytically whether the polynomial function $f(x) = 3x^4 - 2x^2 + 5$ is even, odd, or neither.
Study the parity of the rational function $g(x) = \frac{2x}{x^2 + 1}$, stating whether it exhibits symmetry about the y-axis or about the origin.
Study the parity of the rational function $g(x) = \frac{2x}{x^2 + 1}$, stating whether it exhibits symmetry about the y-axis or about the origin.
Parity in rational and trigonometric functions
Determine analytically the parity of the trigonometric function $f(x) = 2\sin(x)$, recalling the properties of sine for negative angles.
Determine analytically the parity of the trigonometric function $f(x) = 2\sin(x)$, recalling the properties of sine for negative angles.
The profile of an architectural arch is modelled by the function $f(x) = \frac{x^2 - 16}{x^2 + 4}$, where $x$ is the horizontal distance from the centre in metres. Prove…
The profile of an architectural arch is modelled by the function $f(x) = \frac{x^2 - 16}{x^2 + 4}$, where $x$ is the horizontal distance from the centre in metres. Prove analytically whether this profile is symmetrical about the vertical central axis (even function).
Parity of a product of functions
Given two functions, $f(x)$ which is even and $g(x)$ which is odd, prove analytically what type of parity the product function $h(x) = f(x) \cdot g(x)$ has.
Given two functions, $f(x)$ which is even and $g(x)$ which is odd, prove analytically what type of parity the product function $h(x) = f(x) \cdot g(x)$ has.
Study the parity of the function $f(x) = x|x|$, which describes the aerodynamic drag force of a vehicle as a function of its speed $x$.
Study the parity of the function $f(x) = x|x|$, which describes the aerodynamic drag force of a vehicle as a function of its speed $x$.
Parity of a composition
If $f(x)$ is an even function and $g(x)$ is an odd function, determine and rigorously prove the parity of the composite function $h(x) = f(g(x))$.
If $f(x)$ is an even function and $g(x)$ is an odd function, determine and rigorously prove the parity of the composite function $h(x) = f(g(x))$.
Back to the start: If an algorithm penalises a positive deviation and a negative one equally, its error function satisfies $E(-x) = E(x)$: it is an even function, like the absolute error $|x|$ or the squared error $x^2$.