Even and Odd Functions
The main cables of the Golden Gate Bridge hang in a curve that looks the same on both sides of the center of the span. If you stand 500 ft left of center, the cable is at the same height as it is 500 ft to the right. Can we prove that kind of symmetry using only an equation, without drawing anything? That is exactly what the parity of a function tells us.
Even and odd functions: the $f(-x)$ test
Two kinds of symmetry show up again and again. A graph can be a mirror image of itself across the y-axis, or it can look the same after a 180° turn about the origin.
Why the name? The even powers $x^2, x^4, x^6, \dots$ give $(-x)^2 = x^2$, so they are even functions. The odd powers $x, x^3, x^5, \dots$ give $(-x)^3 = -x^3$, so they are odd.
Try it with the sliders. The function is $a x^3 + b x^2$: the $x^3$ part is odd and the $x^2$ part is even. Watch what happens to the symmetry when $a = 0$, when $b = 0$, and when both are nonzero.
For a polynomial there is a shortcut: if every term has an even exponent (a constant counts, since $c = c x^0$), the polynomial is even. If every term has an odd exponent, it is odd. A mix of both kinds is neither.
Analytically determine whether the polynomial function $f(x) = 3x^4 - 2x^2 + 5$ is even, odd or neither.
Analytically determine whether the polynomial function $f(x) = 3x^4 - 2x^2 + 5$ is even, odd or neither.
Prove mathematically whether the polynomial function $h(x) = x^3 - 5x + 2$ has any type of symmetry about the y-axis or about the origin.
Prove mathematically whether the polynomial function $h(x) = x^3 - 5x + 2$ has any type of symmetry about the y-axis or about the origin.
Using values and tables
If you know that $f$ is even or odd, a single value gives you its "twin" for free.
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) = 150…
Knowing that the maintenance cost of a machine follows an even function $C(t)$ with respect to the optimal time $t=0$, and that $\$C(4) = 150$, 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 $p(3) = -12$ ft, what was its position at time $t=-3$ s?
Complete the missing values so that the function $f(x)$ is odd in its entire 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 in its entire domain: $f(1)=4$, $f(2)=-1$, $f(-1)=\dots$, $f(-2)=\dots$
Reading symmetry from a graph
A graph can only suggest symmetry, but it is a fast way to spot it. Use two mental tests.
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Mirror test (even): fold the page along the y-axis. If the left half lands on the right half, the function is even.
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Turn test (odd): rotate the page 180° about the origin. If the graph lands on itself, the function is odd.
When points are marked on a graph, compare the pairs at $x$ and $-x$. Equal $y$-values point to even. Opposite $y$-values point to odd. If some pairs match and others do not, there is no symmetry.
The figure below shows only the right half of a graph (solid). The dashed green curve completes it as an even function. The dashed orange curve completes it as an odd function.
Observe the following graph and classify the represented function as even, odd or with no obvious symmetry, justifying your answer based on the highlighted points.
Observe the following graph and classify the represented function as even, odd or with no obvious symmetry, justifying your answer based on the highlighted points.
The graph shows the right half (for $x \ge 0$) of the profile of a satellite dish. Knowing that its global equation corresponds to an even function, describe what the…
The graph shows the right half (for $x \ge 0$) of the profile of a satellite dish. Knowing that its global equation corresponds to an even function, describe what the left half would be like and what value $f(-x)$ will take as a function of $f(x)$.
Rational and trigonometric functions
The test is the same, but now you must substitute $-x$ in the numerator and the denominator. The parity of each piece tells you what to expect.
Since $\tan x = \dfrac{\sin x}{\cos x}$, we get $\tan(-x) = \dfrac{-\sin x}{\cos x} = -\tan x$, so tangent is odd.
Compare with the next function, which is even. Only even powers of $x$ appear, so substituting $-x$ changes nothing.
Study the parity of the rational function $g(x) = \frac{2x}{x^2 + 1}$, indicating whether it has symmetry about the y-axis or about the origin.
Study the parity of the rational function $g(x) = \frac{2x}{x^2 + 1}$, indicating whether it has symmetry about the y-axis or about the origin.
Analytically determine the parity of the trigonometric function $f(x) = 2\sin(x)$, remembering the properties of the sine for negative angles.
Analytically determine the parity of the trigonometric function $f(x) = 2\sin(x)$, remembering the properties of the 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 to the center in feet. 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 to the center in feet. Prove analytically whether this profile is symmetrical about the central vertical axis (even function).
Absolute value, exponential, logarithmic and radical functions
The test works for every family. You only need to remember how each one behaves when $x$ becomes $-x$.
| Function | $f(-x)$ | Parity |
|---|---|---|
| $\lvert x \rvert$ | $\lvert -x \rvert = \lvert x \rvert$ | even |
| $x\lvert x \rvert$ | $-x\lvert x \rvert$ | odd |
| $3^x + 3^{-x}$ | $3^{-x} + 3^{x}$ | even |
| $3^x - 3^{-x}$ | $3^{-x} - 3^{x}$ | odd |
For logarithms, the key identity is $\ln\dfrac{1}{A} = -\ln A$. When $x \to -x$, a quotient inside a logarithm often flips, and the flip changes the sign of the whole logarithm.
For radical expressions of the form $\sqrt{x^2 + px + q} \pm \sqrt{x^2 - px + q}$, substituting $-x$ swaps the two radicands. The two square roots trade places, so you get either $f(x)$ (with a plus sign) or $-f(x)$ (with a minus sign). Always check that the domain is symmetric too. When the quadratic under the root is always positive, the domain is all real numbers.
A data analysis company uses the function $E(x) = 3^x + 3^{-x}$ to measure the symmetric error of a prediction algorithm. Analyze and prove the parity of this function.
A data analysis company uses the function $E(x) = 3^x + 3^{-x}$ to measure the symmetric error of a prediction algorithm. Analyze 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 in 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 in the interval $(-1, 1)$.
Analyze 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…
Analyze 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 foci located at $F_1\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$ and $F_2\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$.
Products, sums and compositions
You can often find the parity of a combined function without substituting anything, by using the parities of its pieces.
Compositions work in a similar way. Look at what happens inside $h(x) = f(g(x))$ when $x \to -x$.
Compare how the graphs of an even function, an odd function and their product look.
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 $x$.
If $f(x)$ is an even function and $g(x)$ is an odd function, rigorously determine and 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, rigorously determine and prove the parity of the composite function $h(x) = f(g(x))$.
Finding parameters in even polynomials
When a function is required to be even, the equation $f(-x) = f(x)$ must hold for every $x$. Compare coefficients on both sides, or use the fact that an even polynomial has only even powers.
An even polynomial of degree 4 always has the form $f(x) = ax^4 + bx^2 + c$, because only even powers are allowed. That leaves three unknowns, so three conditions determine it.
Calculate the value of the parameter $k$ so that a company's profit function, given by $B(x) = x^2 + (k-3)x + 10$, is an even function and, therefore, symmetrical about…
Calculate the value of the parameter $k$ so that a company's profit function, given by $B(x) = x^2 + (k-3)x + 10$, is an even function and, therefore, symmetrical about $x=0$.
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 complete 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 complete analytical expression of $f(x)$.
Odd rational functions with unknown parameters
Suppose a rational function $F(x) = \dfrac{N(x)}{D(x)}$ has an even denominator (only even powers of $x$). Then $F$ is odd exactly when the numerator $N$ is odd. So all the even-power terms in the numerator must vanish. After that, the remaining conditions (points and asymptotes) give equations for the other parameters.
Two facts you will use:
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A point $(x_0, y_0)$ on the graph gives the equation $F(x_0) = y_0$.
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A vertical asymptote at $x = x_0$ means the denominator is $0$ at $x_0$ (while the numerator is not).
The electromagnetic force of attraction between two metallic components is modeled by an odd function $F(d) = \frac{ad^3 + bd^2 + 5d}{d^2 + 1}$, where $d$ is the…
The electromagnetic force of attraction between two metallic components is modeled by 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 have for this symmetry to be fulfilled.
An engineer designs a mechanical part whose upper contour is given by $f(x) = \frac{ax^3 + cx}{x^2 + b}$. Knowing that the contour is an odd function, passes through the…
An engineer designs a mechanical part whose upper contour is given by $f(x) = \frac{ax^3 + cx}{x^2 + b}$. Knowing that the contour 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 $c$.
Even and odd parts, and derivatives
A surprising fact: every function defined for all real numbers can be split into an even piece and an odd piece.
The exponential function shows the idea beautifully. Take $f(x) = 2^x$ in the graph below (the same method works for $e^x$). The even part is the average of $2^x$ and $2^{-x}$, and the odd part is half their difference. For $f(x) = e^x$ this gives $e^x = \dfrac{e^x + e^{-x}}{2} + \dfrac{e^x - e^{-x}}{2}$.
A look ahead to calculus. The derivative of $f$ at a point is the slope of its tangent line there. Think of an odd function: its graph looks the same after a 180° turn about the origin, and a turn of 180° does not change the slope of a line. So the tangent slopes at $x$ and at $-x$ are equal. In symbols, if $f$ is odd and differentiable, then $f'$ is even.
Prove that any function $f(x)$ defined in $\mathbb{R}$ 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 in $\mathbb{R}$ 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 $I(x) = \frac{f(x)-f(-x)}{2}$.
Applying the principle from the previous exercise, decompose the exponential function $f(x) = e^x$ into the sum of an even function and an odd function.
Applying the principle from the previous exercise, 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 slope of its tangent line at each point. If $f(x)$ is an odd function and differentiable in all of $\mathbb{R}$, prove…
The derivative of a function gives us the slope of its tangent line at each point. If $f(x)$ is an odd function and differentiable in all of $\mathbb{R}$, prove mathematically that its derivative function $f'(x)$ must necessarily be an even function.
Back to the start: a model for the cable height, such as $h(x) = 0.0001x^2 + 220$ (in feet, with $x$ measured from the center of the span), gives $h(-x) = 0.0001(-x)^2 + 220 = h(x)$. The function is even, which proves the left-right symmetry using only algebra.