Parity of a Function
A company in Santiago measures the error of its sales forecasting program symmetrically: being off by $\text{RD}\$1{,}000$ too high or too low counts the same. That symmetry has a name in math: parity. At the end of the page you will see how it is written.
What an Even or Odd Function Is
If you fold a sheet of paper along the $y$-axis and the two halves of the graph coincide, the function is even. If rotating the graph a half turn (180°) around the origin leaves it unchanged, it is odd.
In a table it shows up the same way: if $f$ is even, $f(-3)=f(3)$; if it is odd, $f(-3)=-f(3)$. That is why, in an even model, the maintenance cost at $t=-4$ is the same as at $t=4$; in an odd one, it changes sign.
Knowing that the maintenance cost of a machine follows an even function $C(t)$ with respect to the optimal time $t=0$, and that $C(4) = 9\,000$ pesos, what will be the…
Knowing that the maintenance cost of a machine follows an even function $C(t)$ with respect to the optimal time $t=0$, and that $C(4) = 9\,000$ pesos, what will be the value of $C(-4)$?
The position of an object in a three-dimensional virtual environment is governed by an odd function $p(t)$. If at time $t=3$ s its position on one of the axes is…
The position of an object in a three-dimensional virtual environment is governed by an odd function $p(t)$. If at time $t=3$ s its position on one of the axes is $p(3) = -12$ m, what was its position at time $t=-3$ s?
Complete the missing values so that the function $f(x)$ is odd throughout its domain: $f(1)=4$, $f(2)=-1$, $f(-1)=\dots$, $f(-2)=\dots$
Complete the missing values so that the function $f(x)$ is odd throughout its domain: $f(1)=4$, $f(2)=-1$, $f(-1)=\dots$, $f(-2)=\dots$
Observe the following graph and classify the function shown as even, odd, or without evident symmetry, justifying your answer based on the highlighted points.
Observe the following graph and classify the function shown as even, odd, or without evident symmetry, justifying your answer based on the highlighted points.
Checking Parity with Algebra
The procedure is always the same: substitute $-x$ for $x$ in the whole expression, simplify, and compare with $f(x)$ and with $-f(x)$.
In a polynomial, look at the exponents: if all the terms have even exponents (the constant term counts as $x^0$), it is even; if all are odd, it is odd. If there is a mix, it probably has no parity.
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.
Check mathematically whether the polynomial function $h(x) = x^3 - 5x + 2$ exhibits any type of symmetry about the y-axis or about the origin.
Check mathematically whether the polynomial function $h(x) = x^3 - 5x + 2$ exhibits any type of symmetry about the y-axis or about the origin.
Calculate the value of the parameter $k$ so that the profit function of a company, given by $B(x) = x^2 + (k-3)x + 10$, is an even function and, therefore, symmetrical…
Calculate the value of the parameter $k$ so that the profit function of a company, given by $B(x) = x^2 + (k-3)x + 10$, is an even function and, therefore, symmetrical about $x=0$.
Rational, Trigonometric, Absolute Value, and Exponential Functions
For trigonometric functions, use what you already know about the unit circle: the angle $-x$ is the reflection of $x$ across the horizontal axis, so it keeps the x-coordinate and changes the sign of the y-coordinate.
Also $|-x|=|x|$ (absolute value is even) and $3^{-x}$ is the reciprocal of $3^x$, which will be useful for functions like $3^x+3^{-x}$.
With absolute value, it is sometimes convenient to split into cases. For $f(x)=x|x|$: $f(-x)=(-x)|-x|=-x|x|=-f(x)$, so it is odd.
With logarithms, use $\ln\frac1u=-\ln u$. For example, if $f(x)=\ln\frac{1-x}{1+x}$, then $f(-x)=\ln\frac{1+x}{1-x}=-\ln\frac{1-x}{1+x}=-f(x)$: it is odd.
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.
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 modeled by the function $f(x) = \frac{x^2 - 16}{x^2 + 4}$, where $x$ is the horizontal distance from the center in meters. Prove…
The profile of an architectural arch is modeled by the function $f(x) = \frac{x^2 - 16}{x^2 + 4}$, where $x$ is the horizontal distance from the center in meters. Prove analytically whether this profile is symmetrical about the vertical central axis (even function).
Determine analytically whether the function $g(x) = \frac{x^3 - x}{|x|}$ is even, odd, or neither.
Determine analytically whether the function $g(x) = \frac{x^3 - x}{|x|}$ is even, odd, or neither.
A data analytics firm uses the function $E(x) = 3^x + 3^{-x}$ to measure the symmetrical error of a prediction algorithm. Analyse and prove the parity of this function.
A data analytics firm uses the function $E(x) = 3^x + 3^{-x}$ to measure the symmetrical error of a prediction algorithm. Analyse and prove the parity of this function.
Study the parity of the function $f(x) = \ln\left(\frac{1-x}{1+x}\right)$, used in physics to model certain electric potentials on the interval $(-1, 1)$.
Study the parity of the function $f(x) = \ln\left(\frac{1-x}{1+x}\right)$, used in physics to model certain electric potentials on the interval $(-1, 1)$.
Analyse the parity of the function $f(x) = \sqrt{x^2 + x + 1} - \sqrt{x^2 - x + 1}$, which represents the difference in distances from a point $P(x, 0)$ on the x-axis to…
Analyse the parity of the function $f(x) = \sqrt{x^2 + x + 1} - \sqrt{x^2 - x + 1}$, which represents the difference in distances from a point $P(x, 0)$ on the x-axis to two heat emitting sources located at $F_1\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$ and $F_2\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$.
Graph and Parameters
If you know only half of the graph ($x\ge 0$) and you know the function is even, the left half is the reflection across the $y$-axis, and $f(-x)=f(x)$. If it is odd, it is the reflection through the origin (first across the $y$-axis and then across the $x$-axis), and $f(-x)=-f(x)$.
When the expression has parameters, impose the condition $f(-x)=f(x)$ (or $-f(x)$) and compare the coefficients of each power.
With odd functions, you can also use the fact that the even-exponent terms must disappear: in $F(d)=\dfrac{ad^3+bd^2+5d}{d^2+1}$ the denominator is even, so the numerator must be odd, which forces $b=0$.
The graph shows the right half (for $x \ge 0$) of the profile of a parabolic antenna. Knowing that its overall equation corresponds to an even function, describe what…
The graph shows the right half (for $x \ge 0$) of the profile of a parabolic antenna. Knowing that its overall equation corresponds to an even function, describe what the left half would look like and what value $f(-x)$ will take as a function of $f(x)$.
The electromagnetic attraction force between two metal components is modelled using an odd function $F(d) = \frac{ad^3 + bd^2 + 5d}{d^2 + 1}$, where $d$ is the distance.…
The electromagnetic attraction force between two metal components is modelled using an odd function $F(d) = \frac{ad^3 + bd^2 + 5d}{d^2 + 1}$, where $d$ is the distance. Determine the value that the parameter $b$ must necessarily take for this symmetry to hold.
Let $f(x)$ be a polynomial function of degree 4. If $f(x)$ is an even function, $f(0)=3$, $f(1)=4$ and $f(2)=11$, determine the full analytical expression of $f(x)$.
Let $f(x)$ be a polynomial function of degree 4. If $f(x)$ is an even function, $f(0)=3$, $f(1)=4$ and $f(2)=11$, determine the full analytical expression of $f(x)$.
An engineer designs a mechanical part whose upper outline is given by $f(x) = \frac{ax^3 + cx}{x^2 + b}$. Knowing that the outline is an odd function, passes through the…
An engineer designs a mechanical part whose upper outline is given by $f(x) = \frac{ax^3 + cx}{x^2 + b}$. Knowing that the outline is an odd function, passes through the points $(2, \frac{10}{3})$ and $(3, \frac{15}{4})$, and has a vertical asymptote at $x=1$ (for $x>0$), calculate the exact values of the parameters $a, b$ and $c$.
Product and Composition
Given two functions, $f(x)$ which is even and $g(x)$ which is odd, prove analytically what type of parity the product function $h(x) = f(x) \times g(x)$ has.
Given two functions, $f(x)$ which is even and $g(x)$ which is odd, prove analytically what type of parity the product function $h(x) = f(x) \times g(x)$ 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 $x$.
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 $h(x) = f(g(x))$.
Decomposition into Even Part and Odd Part
Given $f$, define $P(x)=\dfrac{f(x)+f(-x)}{2}$ and $I(x)=\dfrac{f(x)-f(-x)}{2}$.
For derivatives: if $f(-x)=-f(x)$, differentiate both sides (chain rule): $-f'(-x)=-f'(x)$, that is, $f'(-x)=f'(x)$.
Prove that any function $f(x)$ defined on $\mathbb{R}$ can be expressed as the sum of an even function $P(x)$ and an odd function $I(x)$ using the expressions…
Prove that any function $f(x)$ defined on $\mathbb{R}$ can be expressed as the sum of an even function $P(x)$ and an odd function $I(x)$ using the expressions $P(x) = \frac{f(x)+f(-x)}{2}$ and $I(x) = \frac{f(x)-f(-x)}{2}$.
Knowing that any function $f(x)$ defined on $\mathbb{R}$ can be expressed as the sum of an even function $P(x) = \frac{f(x)+f(-x)}{2}$ and an odd function…
Knowing that any function $f(x)$ defined on $\mathbb{R}$ can be expressed as the sum of an even function $P(x) = \frac{f(x)+f(-x)}{2}$ and an odd function $I(x) = \frac{f(x)-f(-x)}{2}$, decompose the exponential function $f(x) = e^x$ into the sum of an even function and an odd function.
The derivative of a function gives us the gradient of its tangent line at each point. If $f(x)$ is an odd and differentiable function on all of $\mathbb{R}$, prove…
The derivative of a function gives us the gradient of its tangent line at each point. If $f(x)$ is an odd and differentiable function on all of $\mathbb{R}$, prove mathematically that its derivative function $f'(x)$ must necessarily be an even function.
Back to the beginning: if a company penalizes an error of excess the same as an error of shortfall, its error function satisfies $E(-x)=E(x)$: it is an even function, like $x^2$ or $|x|$.